The HMI observables are also known as Level-1.5 data, in contrast to Level-0 data (raw HMI images) and the Level-1 filtergrams (Level-0 images at a particular wavelength and polarization that have been corrected for various effects). These observables are separated into two pipelines called LoS and vector magnetic field. The HMI observables are also used to routinely calculate higher-level HMI data-pipeline products that are not described in this article, such as vector magnetic-field maps (Hoeksema et al., 2014), HMI active-region patches (Bobra et al., 2014), synoptic charts and frames, and sub-surface flow maps (Zhao et al., 2012).
The LoS observables are computed from filtergrams taken using the HMI front camera (also called the LoS camera). The LoS observables are images of velocity (Dopplergrams), LoS magnetic field (magnetograms), continuum intensity, and the Fe i line width and line depth. They are produced in two modes: definitive and near-real-time (NRT). The top section of Table 1 lists the Data Record Management System (DRMS) series names computed in the LoS pipeline. All are produced with a 45-second cadence using the front-camera filtergrams, which are observed only in left or right circular polarization.
The vector-field pipeline computes observables using both linearly and circularly polarized filtergrams obtained with the side camera (also called the vector camera). The primary vector-field observable is the set of 24 images comprising the four Stokes-vector elements at each of six wavelengths. The vector-field pipeline runs on a 12-minute cadence and combines filtergrams from ten 135-second sequences. The pipeline also applies the standard LoS observables algorithms to the 720-second Stokes \(I+V\) (RCP) and \(I-V\) (LCP) components to determine averaged LoS quantities. The data series computed in the vector-field pipeline are listed in the lower section of Table 1.
Random uncertainties in the observables are determined largely by the photon noise, which is a feature and consequence of the instrument design. HMI observables meet or exceed the original performance specifications, and estimates of disk-center per-pixel uncertainties due to photon noise are given in Table 1 for each observable. Liu et al. (2012) measured magnetic-field variations in quiet-Sun regions to be no more than 10 % – 20 % larger than the computed photon noise; those observations include unresolved solar signals as well as the effects of other smaller errors due to uncertainties in instrument parameters, such as flat-fielding and shutter noise.
Systematic errors in the observables are larger, more difficult to quantify, and potentially more impactful. They derive from uncertain, irregular, and evolving characteristics of the instrument (e.g. temperature-dependent filter transmission), inherent limitations of the instrument design (e.g. wavelength sampling and resolution), complications of observing a moving, changing Sun (e.g. photospheric evolution during an observation), and environmental conditions (e.g. the effects of the spacecraft orbit). Sources of such errors are a primary subject of this article.
Production of Level-1 Filtergrams
All of the observables are computed from corrected HMI Level-1 images produced as described by Bush et al. (2016). These are stored in the hmi.lev1 (definitive) and hmi.lev1_nrt (near real time: NRT) DRMS data series (see Table 2 and the Appendix for brief definitions of terms). Each record in these series contains two data segments: an image taken by the instrument and a list of bad pixels. Images, referred to as filtergrams, are ordinarily taken at a specific wavelength, i.e. with the instrument filter elements co-tuned. Although not part of the observables processing, it is useful to briefly recall how Level-1 data are obtained from raw images.
Table 2 JSOC, DRMS, and other terminology.
The Level-1 processing makes per-pixel adjustments to the Level-0 data to provide a uniform image for the observable computations. These corrections include removing the CCD overscan rows and columns from the raw Level-0 images, subtracting an offset image to remove the CCD-detector dark current and pedestal, multiplying by a flat-field image to correct gain variations across the detector, and normalizing for exposure time.
The flat fields are monitored and updated weekly. As part of the standard Level-1 flat fielding, pixels with signficantly high or low values are flagged and added to the bad-pixel list that accompanies each Level-1 filtergram. In the definitive data, cosmic-ray pixels in each image are identified based on time-dependent variations from exposure to exposure.
A limb finder determines the nominal coordinates of the solar disk center on the CCD and the observed solar radius. However, because the formation height of the signal changes with wavelength, the radius determined by the limb finder varies as a function of the difference between the target wavelength and the wavelength at the solar limb. At each point, the wavelength shift depends on the changing spacecraft velocity, fixed and variable solar motions and features, and the limb shift. Corrected values are recorded in the crpix1, crpix2, and r_sun keywords, and the reported plate scale is made consistent with the corrected values. The t_obs keyword indicates the center of the time that the shutter was open for each filtergram. exptime is the duration of the exposure. The Level-1 filtergrams are normalized to units of \(\mbox{DN}\,\mbox{s}^{-1}\) by dividing the raw pixel values by exptime.
The production of definitive Level-1 images from Level-0 data may take some time, depending on a variety of factors. For that reason the Level-1 data are produced in two modes: NRT and definitive. NRT observations are intended to be used only for time-sensitive applications, e.g. space-weather forecasting or the development and evaluation of forecasting tools. There are generally only minor differences between the NRT Level-1 images and their definitive counterparts. NRT Level-1 bad-pixel records do not identify cosmic-ray hits because the code computing them requires analysis of a time series. Moreover, the flat fields, daily calibration parameters, drift coefficients, and flight dynamics data applied to the NRT Level-1 images may not be quite up to date. If NRT science data are not retrieved in a timely fashion, records may be skipped. Before definitive Level-1 pipeline processing is completed, the operator manually verifies that all possible data files have been received from the SDO ground system. The complete set of definitive Level-1 records (usually available with a three- to four-day delay) are used for processing the definitive observables. NRT filtergrams and downstream observables are produced within minutes of receiving the raw HMI images.
The website
jsoc.stanford.edu/cvs/JSOC/proj/lev0/apps/build_lev1_hmi.c
provides the source code for the Level-1 pipeline.
The Observables Processing Pipelines
Two distinct software pipelines produce the HMI observables from Level-1 filtergrams. One computes the LoS observables on a 45-second cadence. The other computes time-averaged vector and LoS observables every 720 seconds. Observables are computed from sets of filtergrams taken in a fixed, repeating sequence called a framelist. The framelist specifies the wavelength and polarization state for each exposure. Throughout the prime mission and until 13 April 2016, each camera produced a complete and independent series of filtergrams from which observables have been computed with the corresponding pipeline. The standard 45-second 12-frame sequence used for the LoS camera collects two filtergrams in each of six wavelengths, one in right- and the other in left-circular polarization. The 36-frame vector-camera sequence requires 135 seconds; it collects four additional linear polarization states in each of the six wavelengths to determine the full Stokes vector. The pipelines consist of modules written in C, and the JSOC code is available at
jsoc.stanford.edu/cvs/JSOC/proj
.
The LoS Pipeline – 45-Second Data
The LoS observables calculation is implemented in the HMI_observables module. For each 45-second time step the module identifies and retrieves the proper Level-1 filtergrams, applies a series of corrections to the individual images, interpolates the filtergrams to the specified time, and combines the calibrated filtergrams to compute the observable quantities. Details are provided in the indicated subsections.
Section 2.3 describes how the individual filtergrams are selected and summarizes the calibration processing applied to them. Sections 2.4 and 2.5 give more details about CCD linearity and the corrections for optical, spatial, and temporal alignment. Section 2.7.3 briefly describes the polarization correction made for the LoS magnetic-field computation. Once the filtergrams are fully calibrated, the observables are computed using the MDI-like algorithm described in Section 2.9.
We note that the HMI observables are not cropped precisely at the limb because some provide valid measurements above the limb, e.g. Martínez Oliveros et al. (2014). The crop radius increased from 50 to 90 pixels off the solar limb on 15 January 2014. While the magnetic-field products are just noise off the limb, the project leaves it to the user to choose where to crop depending on the purpose of the investigation.
Section 2.10 describes the current approach implemented to correct the reported velocities for some of the deficiencies caused by SDO’s high orbital velocity using a comparison of the median full-disk velocity to the accurately known spacecraft velocity.
The Vector Observables Pipeline – 720-second IQUV Generation
The vector-pipeline module HMI_IQUV_averaging produces averaged \(I\), \(Q\), \(U\), and \(V\) images at six wavelengths on a regular 12-minute cadence. The vector-field observing sequence, run on the HMI side camera, captures six polarizations at each wavelength according to a repeating 135-second framelist.
The two pipelines share many steps: the gap filling, the de-rotation, the re-centering of the images, and the polarization calibration. However, instead of performing a temporal interpolation, the vector pipeline computes a temporal average. Conceptually, this averaging is executed in two steps. First, like the spatial interpolation described in Section 2.5, a temporal Wiener interpolation of the observed filtergrams onto a regular temporal grid with a cadence of 45 seconds is performed and short temporal gaps are filled. In this case the assumed covariance is derived from the observed average power spectrum of a single pixel in an image. This results in a set of 25 frames for each wavelength/polarization state constructed using the ten original 135-second framelists. The full temporal window over which the interpolation is performed is 1350 seconds, which is wider than the averaging window; a wider window is required because the interpolation needs filtergrams before and after the interpolated times. The 25 interpolated frames are then averaged using an apodized window with a FWHM of 720 seconds; the window is a boxcar with \(\cos^{2}\) apodized edges that nominally has 23 nonzero weights, of which the central nine have weight 1.0 (Hoeksema et al., 2014). In reality the interpolation and averaging are done in a single step for computational efficiency.
Following the temporal averaging, the six polarized filtergram are converted into a Stokes [IQUV] vector. The details of the polarization calibration are given in Section 2.7. The final results are stored in hmi.S_720s. The other 720-second observables are computed from \(I \pm V\) using the same algorithm as the LoS pipeline (Section 2.9).
Filtergram Selection, Mapping, and Image Processing
This section describes how filtergrams for the LoS pipeline are selected and reviews how they are processed individually to correct for various problems. Subsequent sections detail how the pipeline deals with nonlinearity in the CCD cameras (Section 2.4) and with spatial alignment and distortion corrections (Section 2.5). Section 2.6 describes the relative and absolute roll angles of the CCD cameras and validation of the distortion and roll determination using the Venus transit. Later sections describe issues with polarization (Section 2.7) and the wavelength filters (Section 2.8).
The filtergram-selection module gathers selected keywords from all of the Level-1 records in a time interval around the target time [t_obs] at which an observables computation is requested by the user. The Level-1 image at the proper wavelength nearest to the target time for which the observables record is to be produced is identified as the target filtergram. Since launch, the HMI filtergram closest to and slightly blueward of the rest wavelength of the core of the Fe i line has been used. Certain keywords of the target filtergram are used as reference values for the final observable data record. For instance, the focus block used to take the target filtergram is the reference focus block for the computation. If another filtergram is taken with a different focus, this is an error and no observable will be created.
The code then locates another filtergram taken with the same wavelength and polarization settings as the target to linearly interpolate values of the components of the spacecraft velocity [obs_vr, obs_vw, and obs_vn], the solar distance [dsun_obs], the Carrington coordinates of the disk center [crlt_obs and crln_obs], and the position angle of the Carrington rotation axis [crota2] for the target time t_obs. Spacecraft ranging data regularly confirm that the location of the spacecraft is known to [much] better than 120-m accuracy and the velocity to better than \(0.01~\mbox{m}\,\mbox{s}^{-1}\). We note that crota2 is the negative of the classic p-angle. As SDO has been operated since launch, the HMI crota2 has remained close to −180 degrees except during roll maneuvers (see Section 3.1).
A gap-filling routine is called to replace the pixels identified in the bad-pixel list. Similar to the spatial interpolation (see Section 2.5 for details), the gap-filling uses a Wiener interpolation assuming the covariance function obtained from the perfect modulation transfer function (MTF). However, unlike that interpolation, this is done as a general 2D interpolation, with weights calculated based on the exact pattern of missing pixels surrounding the target pixel. Furthermore, a term is added to the optimization to minimize both the sum of the variance from the inaccurate interpolation and the variance from the photon noise, rather than just the former.
The HMI cameras are affected by a small nonlinearity in their response to light exposure that is corrected separately for the LoS and vector cameras (see Section 2.4).
The code then retrieves and corrects each of the Level-1 filtergrams needed to produce an image at time t_obs with the required wavelength and polarization setting. When all of the necessary filtergrams have been prepared, an interpolation module performs several tasks. First it corrects each image for instrumental distortion. The distortion as a function of position is reconstructed from Zernike polynomials determined during pre-launch calibration that employed a random-dot target mounted in the stimulus telescope (Wachter et al., 2012). Section 2.5 describes an evaluation of the measured instrumental distortion. The routine also corrects the Sun-center coordinates and solar-radius keywords to account for modifications due to the distortion correction. Because each Level-1 record used to compute an observable is taken at a slightly different time, features on the solar disk move a small fraction of a CCD pixel, so that a given pixel in two filtergrams does not sense exactly the same location on the solar surface. This rotation is corrected to subpixel accuracy using a Wiener spatial-interpolation scheme. The time difference used to calculate the pixel shift is the precise observation time of the Level-1-filtergram t_obs. The filtergrams are also re-centered and re-sized to the same values. These values are obtained by averaging the characteristics of all the Level-1 images used to produce the observable. Next the filtergrams are temporally interpolated to the target observable time t_obs. For a given wavelength/polarization pair, the temporal interpolation ordinarily requires six Level-1 filtergrams in definitive mode and two in NRT mode to interpolate to the requested time. The two-point NRT temporal interpolation is a basic linear scheme, while the definitive six-point method uses the specific weighting scheme described in Martínez Oliveros et al. (2011). When the loop over all wavelengths and polarizations is complete, the result is a set of filtergrams with the same solar radius and Sun-center position all interpolated to the proper observable time: t_obs at the spacecraft.
We note that the observables have at least two time keywords: t_rec and t_obs. t_rec is a prime keyword and is the time at which the data would have been observed at precisely 1 AU, whereas t_obs is the clock time of the midpoint of the observation at the SDO spacecraft: t_obs = t_rec + (dsun_obs - 1 AU)/c, where c is the speed of light. The t_rec times are by design uniformly spaced in (TAI) time and therefore convenient to use. However, the time of observation for which the orbit keywords best describe the observable is the time at SDO: t_obs.
The final processing step is to correct for polarization issues and create a set of filtergrams that better represent the true \(I \pm V\) polarizations with less cross-contamination; those images are called Level-1p data. See Section 2.7.3 for further discussion. The Level-1p records can optionally be saved.
CCD Nonlinearity
The signal (in DN) measured in a given pixel does not vary exactly linearly with the number of incident photons. Wachter et al. (2012) determined that the nonlinearity of the HMI CCDs is on the order of 1 % for intensitites lower than 12,000 DN from ground-calibration data. Typical 140 ms exposures are about 4200 DN, giving median normalized Level-1 filtergram values of 30,000 – \(50{,}000~\mbox{DN}\,\mbox{s}^{-1}\), depending on the wavelength. Even apart from solar variations, the number of photons received by each CCD pixel is not constant for a given exposure time for a variety of reasons, including the daily change in the Sun–SDO distance. A nonlinearity correction is therefore implemented in the observables pipelines for each Level-1 image.
The initial correction was based on the results of Wachter et al. (2012). The difference between the actual intensity and a linear response was fit as a function of intensity using a third-order polynomial. The coefficients of this polynomial were −11.08, 0.0174, \(-2.716 \times 10^{-6}\), and \(6.923 \times 10^{-11}\) for the front camera and −8.28, 0.0177, \(-3.716 \times 10^{-6}\), and \(9.014 \times 10^{-11}\) for the side camera.
Since 15 January 2014, different coefficients have been used. The main reason for the change is that the negative value for the zeroth-order coefficient (intercept term) in the original fits means that a few pixels ended up with a negative (albeit small) intensity, which does not make physical sense. The coefficients used after 15 January 2014 are 0, 0.0207, \(-3.187 \times 10^{-6}\), and \(8.754 \times 10^{-11}\) for the front camera and 0, 0.0254, \(-4.009 \times 10^{-6}\), and \(1.061 \times 10^{-10}\) for the side camera.
The version of the nonlinearity calibration is given in the keyword calver64; see Section 2.8.2.
Calibration sequences are taken regularly on-orbit to monitor the nonlinearity of the CCDs. Although a slightly different nonlinear response can be detected in each quadrant (Figure 19 of Wachter et al., 2012, shows a total spread of no more than 5–10 DN), we measure the spatial average over the entire CCD. So far, the nonlinearity has proven constant. Figure 1 shows the result for a typical nonlinearity sequence recorded on 16 October 2013.
Distortion Correction and Image Alignment
Image distortion arises because of small imperfections in the optics, including the optics that move to tune the instrument. The correction is based on Zernike polynomial coefficients measured with ground data taken prior to launch (Figure 7 of Wachter et al., 2012). The mean residual distortion is \(0.043 \pm 0.005\) pixels, with a maximum smaller than 2 pixels near the top and bottom of the CCD camera. Differences between the cameras are of order 0.2 pixels. The instrumental-distortion correction is applied to each Level-1 filtergram. Production of definitive LoS observables typically involves 72 filtergrams, while a definitive 12-minute averaged Stokes vector requires 360 Level-1 filtergrams.
Each filtergram has slightly different Sun-center location coordinates and p-angle. Therefore, before performing a temporal interpolation to t_obs, each Level-1 filtergram must be registered and aligned. Conceptually, each image is first rotated to a common p-angle and adjusted to a common solar inclination (\(\mathrm{B}_{0}\)-angle). Then the effect of solar differential rotation is removed by spatially interpolating to the proper spatial coordinates at the target time. (Near disk center, solar rotation carries features across a pixel in about three minutes.) The de-rotated images are then re-centered around a common Sun-center point that is the average of all the input Level-1 filtergrams. In practice, all of these operations are performed in a single interpolation step.
The spatial interpolation is made using a separable (in \(x\) and \(y\)) Wiener interpolation scheme of order ten. Such a scheme minimizes the rms error of the interpolation for a specified covariance. Here the covariance function corresponds to the ideal diffraction-limited MTF of the instrument, i.e. the covariance expected if a white-noise signal were observed by an ideal instrument. Of course, the instrument is not ideal and the inherent solar signal is not white, but this only affects the relative convergence at different spatial frequencies. Because the HMI pixels undersample the point spread function (PSF) by about 10 %, it is impossible to perfectly interpolate the data. However, this only affects the spatial frequencies to which the super-Nyquist signal folds. Consequently, the interpolation (and the resulting power spectrum) is very good up to \({\approx}\,0.9\) times the Nyquist frequency and imperfect above that.
After the image distortion has been corrected, an analytical correction is applied to the original Level-1 Sun-center position and solar radius determined by the limb finder. Unfortunately, tests show that the analytical correction is not precisely correct, particularly for the \(y\)-direction Sun-center location reported in keyword crpix2 (nominally the solar north–south location). The crpix2 value is systematically greater than the more accurate center position determined when the limb finder is applied to the distortion-corrected image by 0.13 to 0.17 pixels, depending on the filtergram wavelength. The systematic \(y\)-axis difference also depends predictably on image location. In the \(x\)-direction (reported in crpix1) the center-position difference is roughly ten times smaller, and for the solar radius it is less than a hundredth of a pixel. Consequently, users are cautioned that currently the crpix2 keyword in the observables records is systematically off by \({\approx}\, 0.15\) pixels. Random variations in measured center position from one image to the next due to all noise sources are \({<}\, 0.15\) pixels for the 45-second observables and \({<}\, 0.05\) pixels for the 720-second observables.
Roll, Absolute Roll Calibration, Distortion, and the Venus Transit
This section describes the determination of the relative roll angle of the two HMI cameras from daily calibration measurements and how the roll difference varies with time. Analysis of Venus-transit observations provides an accurate determination of the absolute roll angle of the HMI instrument, as well as independent information about optical distortion. Knowledge of the instrument roll angle is important because solar rotation can be transferred into measurements of north–south motions; for example, a roll error of \(0.1^{\circ}\) would introduce a systematic \(3.5~\mbox{m}\,\mbox{s}^{-1}\) northward or southward flow. Investigations that depend on coalignment of filtergrams or with measurements from other instruments can also be affected.
The first objective is to determine the difference in the roll between the two cameras. Pairs of adjacent (in time) images on the front and side cameras with otherwise identical settings are corrected for distortion using the same parameters employed to make the observables. The images are then divided into \(256 \times 256\)-pixel regions, each of which is high-pass filtered and circularly apodized with a raised cosine between \(0.8 \times 128\) pixels and \(0.9 \times 128\) pixels. For each region the two images are then cross correlated to determine the shift in \(x\) and in \(y\). Finally, for locations inside the solar image, these shifts as a function of position on the image are fitted to a model of the shifts with parameters for the \(x\)- and \(y\)-offsets, a scale error, and a roll angle.
First this was applied to all 640 pairs of images with one particular polarization and wavelength setting on 1 July 2012, which gave a mean offset of \(0.08361^{\circ}\) with a scatter of \(0.00061^{\circ}\) and thus a resulting error of the mean of \(0.00002^{\circ}\). An estimate of the accuracy of the number is difficult to obtain, but the rms residual in each direction is roughly 0.1 pixels and, as shown in the next subsection, the distortion model is most likely uncertain by about the same amount. Since the average solar radius is \({\approx}\, 1900\) pixels, it is reasonable to expect an uncertainty of about 0.1/1900 radians or \(0.003^{\circ}\). Tests performed on a few other days gave average roll offsets that are slightly higher: about \(0.08376^{\circ}\). A nominal value of \(0.0837 ^{\circ}\) has been adopted for the roll difference between the two cameras.
As in the case of other datasets, the intermediate data series were not saved, but a rerun of the code on images with current calibration and processing gives values consistent to the number of significant digits given above.
To investigate the time dependence of the roll difference, the process is repeated on the daily calibration images taken by HMI, specifically on the pairs of tuned continuum images obtained with the two cameras at about 06:00 and 18:00 UT. Figure 2 shows that the roll difference drifts slowly with time at a uniform rate of \(-0.00020^{\circ}~\mbox{yr}^{-1} \pm 0.00006^{\circ}~\mbox{yr}^{-1}\). This corresponds to an offset of roughly 0.03 pixels at the limb over the five years, which may be significant when one attempts to combine data from the two cameras. While the cause of the small drift is not known, three possibly relevant improvements to the instrument temperature control scheme were made that may have decreased the scatter in the measurement: one on 16 July 2013 (mission day 1142) to the optical bench, another to the telescope tube on 25 February 2014 (day 1395), and the third on 16 and 26 June 2014 (days 1517 and 1527) to the front window (see Bush et al.
2016 for details.)
The Venus transit of 5 – 6 June 2012 provided a unique opportunity to test the accuracy of the absolute roll angle and parts of the distortion map. The normal filtergram sequence was run on the front camera, but on the side-camera images were taken in the true continuum in linear polarization. Images first undistorted in the standard way have a simple model of the solar limb darkening removed. Then an area of roughly \(100 \times 100\) arcsec around the expected position of Venus is extracted from each image and the radial derivative of the intensity is calculated. This is then multiplied by an apodization function to isolate the limb, cross correlated with a \(180 ^{\circ}\)-rotated version, and the center position of the Venus image is determined.
The Venus positions are then fit, separately for each camera, to the ephemeris using a model that determines the image offset, image scale, and roll angle. Considering the accuracy of the spacecraft orbit, the ephemeris is presumed to be perfect. The small roll-angle values reported in this section are the p-angle relative to the nominal orientation of the instrument, i.e. p-angle = 180 − crota2, with solar South at the top of the CCD. The original analysis determined absolute roll angles of \(-0.0142 ^{\circ}\) and \(+0.0709 ^{\circ}\) for the two cameras, with the difference being \(0.0851 ^{\circ}\). A reanalysis of the same images processed with current calibration software gives roll angles of \(-0.0140 ^{\circ}\) and \(+0.0712 ^{\circ}\), corresponding to a roll difference of \(0.0852 ^{\circ}\). Compared with the value determined by direct comparison reported in the previous section (\(0.0837 ^{\circ}\)), this confirms that the absolute roll values are likely accurate to a few thousandths of a degree.
Residuals in \(x\) and \(y\) for each camera are shown in Figure 3. The distortion-corrected data (black points) show substantially reduced residuals, but systematic deviations of about 0.1 pixels still remain. This is consistent with roll uncertainties of about \(0.003 ^{\circ}\), confirming that the discrepancies between the methods are negligible.
A second analysis using the Venus transit was performed to better characterize the image distortion of the side camera. A PSF estimate (see Section 3.6) for HMI was deconvolved from the images using a Richardson–Lucy algorithm to improve the estimates of the Venus-center location. The results for roll and distortion are not significantly different.
The similarities in the results for the front and side cameras shown in Figure 3 suggest that the remaining distortion errors arise in parts of the optical path common to both cameras. The increased scatter (even sometimes double values) in the residuals for the front camera shown in the lower panels of Figure 3 were due to filtergrams taken in different wavelengths, so there is a remaining sensitivity to tuning at the level of 0.03 pixels. This confirms that the instrumental-distortion model is accurate to about 0.1 pixel, at least for the pixels lying on the Venus path.
Finally, the two sets of numbers must be reconciled. The roll difference from the direct comparison (\(0.0837 ^{\circ}\)) differs from the Venus data (\(0.0851 ^{\circ}\)) by less than the expected uncertainty. Furthermore, the Venus numbers are effectively derived from only a small portion of the solar images, whereas the direct comparison is effectively an average over the full disk. Since one of the objectives is to be able to combine full-disk data from the two cameras, it was decided to use the direct-comparison difference and the well-determined absolute average p-angle from the Venus measurements. For the two cameras we obtain the results given in Table 3.
Clearly, it is desirable to obtain independent determinations of the absolute roll. While no other planetary transits occurred during the prime phase of the mission, we plan to analyze observations of the Mercury transits on 9 May 2016 and 11 November 2019. Use of more frequent lunar transits presents significant difficulties. Only the ingress can be used because of the massive thermal perturbations later in the transits. Complications also arise because the Moon is far from perfectly spherical. Lunar mountains are clearly visible in the HMI images, so accurate maps of the lunar topography would have to be used to derive an estimate. An attempt to detect the star Regulus was made, but was not successful.
Polarization
This section describes some of the issues with telescope polarization and corrections made to minimize contamination. The pipeline module polcal performs the calibration in several steps.
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First the model described by Schou et al. (2012b) is used to determine the modulation matrix for the frames. This takes into account the polarization-selector position and temperature. Because the temperature gradient across the front window is not known, it is assumed to be zero. A single average temperature is used for all frames. Given the nature of the model, the \(I\) → (QUV) and (QUV) → I terms used here are zero and the \(I \rightarrow I\) term is unity. Because the calibration model is only given on a \(32 \times 32\) grid, so is the modulation matrix at this point.
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At each pixel a least-squares fit is performed to determine the demodulation matrix, which relates the observed intensities in the various frames to \(I\), \(Q\), \(U\), and \(V\).
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For each pixel in the \(4096 \times 4096\) image the demodulation matrix is linearly interpolated from the \(32 \times 32\) grid and applied to the observed intensities.
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A telescope polarization correction is applied by subtracting a small part of \(I\) from \(Q\), \(U\), and \(V\). The model used and how it was determined is described in Section 2.7.1.
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A polarization-PSF correction is made by convolving \(I\) with \(5 \times 5\) kernels and adding the result to \(Q\), \(U\), and \(V\). The model used and how it was determined is described in Section 2.7.2.
Calibration of the circular polarization measured with the front camera is briefly described in Section 2.7.3.
The calibration requires the temperatures of some HMI components that impact the polarization calibration. The temperatures are updated once a day for the entire day. In NRT mode, default temperature values are used.
Telescope Polarization
The determination of the telescope polarization was performed after launch because it is notoriously difficult to determine accurately on the ground and quite straightforward to determine on orbit.
To determine the telescope-polarization term, 820 12-minute IQUV-averages with good quality from nine days between 3 May 2010 and 3 September 2010 were used. The images of \(Q\), \(U\), \(V\) were first binned to \(256 \times 256\) pixels and divided by the corresponding binned \(I\)-images; then for each wavelength, polarization, and pixel the median-in-time over the 820 samples was calculated. The median was used to suppress the effect of solar activity. Because activity is still quite visible in the line and because using the continuum is in any case preferable, the two wavelength positions closest to the continuum (I0 and I5) were averaged, which also has the effect of canceling the effects of the orbital-velocity variations to lowest order.
The resulting images are shown in the top row of Figure 4. The dominant effect is an offset, but there is also a radial gradient. A fourth-order polynomial in the square of the distance to the center of the image models this well. Since the effect in \(V\) is small, no correction is applied to it. The bottom row in Figure 4 shows the residual after the polynomial has been subtracted. While the constant part could indeed be due to a small amount of polarization in some optical component, the cause of the radial variation is unknown. However, the variation is very similar for \(Q\) and \(U\). The rms residuals within a distance of 0.85 times the half width of the image (corresponding roughly to 0.93 times the radius of the solar image) are 14 ppm, 28 ppm, and 18 ppm for \(Q, U\), and \(V\), respectively. The most prominent features in the residuals are the arcs on the left. The cause of the arcs is unknown, but they may be due to a ghost reflection.
The instrument easily meets the original specification for polarization, which is 1000 ppm for \(I\rightarrow QUV\) and 100 ppm for cross-talk among \(Q\), \(U\), and \(V\) (Schou et al., 2012b).
While the correction is modeled as if it originated in the telescope part of the instrument, it is clear that it actually does not. During commissioning, a different set of polarization settings was used and the constant terms were quite different. As such it is likely that the root cause lies elsewhere, such as in second-order effects in the waveplates, in the fold mirror, or in the polarizing beamsplitter; however, this has not been investigated further.
Given this lack of physical understanding, the term has not been used to determine the corresponding (\(QUV\)) → \(I\) terms, which are left at zero. Given that the terms are on the order of \(10^{-4}\), the effect on the intensity term is in any case very small.
Some of the files used for the original calibration are no longer available and reproducing them with exactly the same calibrations and corrections is not practical. To verify that the results are reliable, the analysis was repeated on the previously calibrated data using the same set of 820 12-minute averages. Ideally, this should result in zero for the calibration terms. In reality, it leads to polynomials that are nearly constant across the disk with offsets for \(Q\) and \(U\) of \(-1.4 \times 10^{-6}\) and \(1.1 \times 10^{-6}\). For \(V\) (which does not have a calibration applied) the mean correction is \(-1.6 \times 10^{-9}\). The cause of the small change is unknown, but one of the differences in processing is that bad pixels may have been unfilled in the original analysis, and a second is that the polarization-PSF correction described below was not applied. In any case, the differences are negligible.
To determine whether the correction is stable in time, 12-minute averages with perfect quality for the first day of each month from May 2010 through August 2015 are analyzed in the same way. Figure 5 shows the mean of the telescope-polarization terms. The top panel shows \(I\) → \(Q\), the middle panel \(I\) → \(U\), and the bottom panel \(I\) → \(V\), in parts per million. Some temporal changes are seen, with a magnitude of up to at most \({\approx}\, 2 \times 10^{-5}\) with some annual periodicity. The cause of these changes is unknown, but they could be due to changes in the front-window temperature. In any case, the changes are small compared to the residuals shown in Figure 4 and the correction has been kept fixed in time.
Polarization-PSF Correction
After the telescope-polarization correction is applied, another artifact becomes apparent, viz. a granulation-like pattern appearing in \(Q\) and \(U\) (but again not in \(V\)). An analysis of the average power spectrum of the \(I\), \(Q\), \(U\), and \(V\) components (Figure 6) shows that the pattern is not simply a leak of \(I\) into \(Q\) and \(U\), but it appears instead to be a filtered version of \(I\).
While the instrumental cause is unknown, the spectra look similar to what might be expected if there were a different PSF when observing, for example, \(I+Q\) and \(I-Q\). To correct for the effect, a least-squares minimization procedure (including a small amount of regularization) is used to find the best \(5 \times 5\) kernel that, when convolved with \(I\), best reproduces \(Q\), \(U\), and \(V\). We impose the additional mathematical constraint that the sum of the kernel is zero to avoid interference with the telescope polarization correction. This estimate is computed with a subset of 612 samples from the same datasets used for the telescope polarization. From these a \(512 \times 512\) patch at disk center is extracted and only the two continuum-wavelength polarizations are used. As can be seen from the bottom panels of Figure 6, this procedure dramatically reduces the contamination. Larger sized kernels and spatially varying kernels were also evaluated, but they did not substantially improve the results.
Because the datasets as processed for the original analysis are no longer available, new ones had to be constructed using images computed using the current calibration to verify the reproducability of the result. This was done by convolving \(I\) with the adopted kernels and adding it back into to the already corrected \(Q\)- and \(U\)-images before repeating the analysis. The results are very nearly, but not exactly, identical to the original. The differences are much smaller than the photon noise and do not cause noticeable differences.
The polarization-PSF correction has been estimated over the entire mission. Results are shown in Figure 7. Here it is, unfortunately, the case that there are substantial variations with time. The cause of this effect is not understood either. The effect varies with polarization-selector setting, and once again, the likely causes include second-order effects in the waveplates, the fold mirror, and the polarizing beamsplitter. It is interesting that the short-term temporal variations increase about the same time that the thermal-control scheme for the optical bench changed.
LoS Polarization Contamination
For the front camera, where only LCP and RCP are observed, it is not possible to perform a full demodulation. Instead, it is assumed that only \(I\) and \(V\) are non-zero, and only these two are inverted for. In reality, \(Q\) and \(U\) are non-zero and will leak into the resulting \(I\) and \(V\). Given the existing polarization model, it is straightforward to calculate these leaks, and the results are shown in Figure 8. Indeed, among the many possible polarization-selector settings, those with small leaks were selected when the framelists were created. The terms are quite small and much better than the original specification of 5 % for leakage.
Filters
The HMI filter system consists of the entrance window, a broad-band blocking filter, a multi-stage Lyot filter, the last element of which is tunable, and two tunable Michelson interferometers. The following sub-sections describe the detection and mitigation of wavelength non-uniformities determined for the filter elements in phase maps (Section 2.8.1), the correction for interference fringes created by the front window that are most visible when the instrument is operated in calibration mode (CalMode: Section 2.8.2), and a tuning-dependent intensity variation (\(I\)-ripple), created by imperfections in the tunable-filter elements, that leaks into the look-up tables (Section 2.8.3).
Phase Maps: Filter-Element Wavelength Non-Uniformity
Ideally, the wavelength would be the same at each point of a filtergram, but in practice, the filter elements are not completely uniform and drift with time. Thus the actual wavelength at each pixel depends on the ray path through the filter elements in the instrument. We characterize these imperfections as phase maps.
The tunable elements of HMI – the two Michelsons and the narrowest Lyot element (E1) – have a wavelength-dependent transmittance [\(T(\lambda)\)] that is modeled as
$$ T(\lambda) = \frac{1+ B \cos(2 \pi \lambda/\mathrm{FSR} + \Phi + 4 \phi) }{2}, $$
(1)
where FSR is the free spectral range of the element, \(\phi\) is the phase resulting from the tuning motor position, and \(\Phi\) is the phase due to the properties of the filter. The \(\Phi\) for each element can be determined by tuning each filter separately in what is called a detune sequence. Detune sequences are currently obtained every other week to make spatially resolved maps of \(\Phi\) for all three tunable elements for each CCD (Couvidat et al., 2012b). These phase maps are used to determine the effective wavelength at each location and thus to provide a spatially dependent calibration of the look-up tables used to determine the observables with the MDI-like method (described in Section 2.9). Because the variations across the filters have low spatial frequency, phase maps are computed and stored on a \(128 \times 128\) grid and interpolated to the full resolution of the CCDs when used. Figure 9 shows the phase maps for the tunable filter elements. The narrow-band Michelson is the most sensitive to tuning phase, \({\approx}\, 31~\mbox{m}\,\mbox{s}^{-1}\) per degree of phase. The CalMode fringe correction is described in Section 2.8.2.
The spatially dependent phase-map patterns are relatively constant in time for all three filter elements; however, the central wavelengths do drift. The narrow-band Michelson drifts at \({\approx}\, 6~\mbox{m}\mathring{\mathrm{A}}\) per year and has a small annual periodicity. The wide-band Michelson drift is \({\approx}\, 30~\mbox{m}\mathring{\mathrm{A}}\) per year and is slowing with time. Both contribute about the same amount to an annual shift in the wavelength zero point that corresponds to a velocity shift of \({\approx}\, 100\) – \(200~\mbox{m}\,\mbox{s}^{-1}\). The drift is such that the elements remain fairly well co-tuned for extended intervals. The broader E1 Lyot element drifts more slowly: \({<}\, 7~\mbox{m}\mathring{\mathrm{A}}\) per year. The observables look-up tables are updated approximately once per year to account for the drifts when the velocity offset exceeds a couple hundred \(\mbox{m}\,\mbox{s}^{-1}\) (for dates see Table 5 in Section 2.10).
It was noticed early on that the average phases of the tunable elements differ slightly between front and side cameras. The largest difference is \({\approx}\, +0.55^{\circ}\) for the narrow-band Michelson, and it exhibits a very slow increase with time (\({\approx}\, 4.3 \times 10^{-5}\) degrees per day). The broad-band Michelson and Lyot E1 filter elements have smaller phase differences of about \(-0.1^{\circ}\) and \(-0.13^{\circ}\) that appear stable in time. These front- and side-camera phase differences most likely originate in the polarizer of the narrow-band Michelson. A small leak of the orthogonal polarization will be picked up by the polarizing beam-splitter used to separate the two light paths.
Removal of CalMode Fringes from the Phase Maps
The phase maps of the tunable elements are determined using detune sequences made in calibration mode. In CalMode, calibration lenses are inserted into the HMI light path to switch conjugate planes in the optical path. This puts an image of the entrance pupil onto the CCD, rather than an image of the Sun. The purpose of CalMode is to provide uniform disk-average line profiles all across the CCD field, thus eliminating solar spatial information as well as the effects of having different angular distributions of rays through the filter section.
The front window of HMI is constructed of multiple layers of different glass and glue (with different refractive indices); therefore it acts as a weak Fabry–Perot interferometer. In CalMode the fringes from the front window, and to a lesser degree the blocking filter, are imaged onto the CCD. Consequently, the phase maps of the tunable elements also show the interference fringes. The front-window fringes are not present in the regular observing mode (Obsmode) images, but because the phase maps are used to derive the look-up tables for the MDI-like algorithm (Section 2.9.2), the CalMode interference fringes bleed into LoS observables. For helioseismic purposes this is mainly a cosmetic issue that has minimal scientific impact on the determination of frequencies. On the other hand, it can adversely impact secondary objectives, such as determination of surface flows directly from the Doppler shift.
Biweekly detune sequences are used to compute (uncorrected) phase maps. Each phase map contains five \(128 \times 128\) images: one for each of the three tunable filter elements, and two more for the line-width and line-depth maps of the Fe i line fitted together with the phases. The method described below mostly removes the obvious effects of the fringes present in the CalMode phase maps, but the user is cautioned to remember that this does not necessarily mean that the inferred phase values for the CalMode fringes are actually correct. A new correction is computed each time we retune the HMI instrument or change the line or filter calibration (see Table 5 in Section 2.10), since retuning requires computing new look-up tables for the MDI-like algorithm. Finally, it is noteworthy that the CalMode fringe correction has only been made to look-up tables used to compute observables collected from 1 October 2012 onwards. Earlier observables have not be reprocessed, as indicated in the calver64 keyword (see
jsoc.stanford.edu/jsocwiki/CalibrationVersions
for details). For this reason, earlier observables computed using the phase maps still show the fringe pattern.
To model the fringes we start by assuming that each of the five phase maps can be written as
$$ A(x,y,t)=A_{0} (x,y)+A_{\mathrm{L}}(x,y) t + A_{\mathrm{C}} (x,y) \cos\bigl(\phi(t)\bigr) + A_{\mathrm{S}} (x,y) \sin\bigl(\phi(t)\bigr) , $$
(2)
where \(A_{0}\) represents a constant term; \(A_{\mathrm{L}}\) is a term describing the overall drift; \(A_{\mathrm{C}}\) and \(A_{\mathrm{S}}\) describe the fringes, all arbitrary functions of space; and \(\phi\) describes the phase of the fringes (effectively the glass thickness) as a function of time. The fringes are most cleanly visible in line depth, so we start by fitting Equation (2) to that. The fit is performed using the terms of a singular value decomposition (SVD) as the initial guess and alternately fitting \(\phi\) and the spatial terms.
Once \(\phi\) is determined, Equation (2) is fit for each of the five variables and an estimate of the corrected Obsmode phase map is determined by subtracting the fringe term \(A_{\mathrm{C}} (x,y) \cos(\phi(t))+ A_{\mathrm{S}} (x,y) \sin(\phi(t))\) from the original values.
Clearly, modeling the fringes like this is far from perfect, but the result is nonetheless that the amplitude of the large-scale fringes is dramatically reduced. Unfortunately, this leaves behind a number of smaller-scale fringes. Repeating the above procedure on the residuals (replacing the \(A\)-terms with equivalent \(B\)-terms and \(\phi\) with \(\phi_{1}\)) to remove these is not nearly as efficient as for the larger fringes, but does nonetheless improve the results significantly, therefore both corrections are applied.
Figure 10 shows the \(A_{\mathrm{C}}\), \(A_{\mathrm{S}}\), \(B_{\mathrm{C}}\), and \(B_{\mathrm{S}}\) terms for line depth. The \(A\)-terms capture large-scale concentric fringes, parts of the oval-shaped medium-scale features above and to the left of image center, and some of the strong small-scale pattern. The \(B\)-terms capture the stronger small-scale fringes most prominent near the top of the image and more of the oval-shaped features. Figure 9 shows the impact of the correction on the tuneable-element phase maps: the left panels show the raw maps, while the right panels show the same phase maps after the fits to the CalMode interference fringes have been removed. The CalMode fringe corrections are as much as 25 % the magnitude of the of the corrected phase maps, e.g.
\({\pm}\, 2.5^{\circ}\) in the narrow-band Michelson, but with a very different spatial scale. While the phase patterns intrinsic to the elements remain, it appears that the CalMode front-window fringes have been mostly removed. The intense small-scale fringes near the top of the circle are most obvious in the narrow-band Michelson and have been dramatically reduced in amplitude. The roughly concentric pattern is significantly reduced in both Michelsons, although it is less obviously present in the Lyot. Unfortunately, the oval-shaped medium-scale features bleed through in both Michelsons, but not as much in the Lyot. Small-scale fringes not due to the CalMode fitting are also present. It is interesting to note that \(\phi_{1} \approx 2.25~\phi\). Given the thicknesses of the different glass elements (6 mm, 3 mm, and 6 mm in the order traversed), a factor of 2.00 or 2.50 might naively be expected, the deviation presumably being due to the different thermal-expansion coefficients and/or different changes in the refractive index with temperature.
More elaborate correction schemes were also investigated, but they did not provide a significant improvement.
\(I\)-Ripple Characterization
What we call \(I\)-ripple is an intensity variation in the HMI output that depends on the instrument tuning. \(I\)-ripple results from imperfections in the tunable-filter elements, such as small misalignments in wave-plates. It is most apparent when using a uniform and constant light source. No correction is currently made for \(I\)-ripple when deriving the filter-transmission profiles. This introduces a small error in these profiles and therefore in the look-up tables used by the MDI-like algorithm. Simulations show that \(I\)-ripple produces a systematic zero-point error in velocity of at most a few tens of meters per second that varies linearly with Sun–SDO velocity. For details of the derivation and the results of HMI ground calibrations, see Couvidat et al. (2012b) and references therein.
For a specific tuning phase [\(\phi\)] of the tunable Lyot element [E1], the transmitted intensity can be modeled as
$$ \frac{I(\lambda)}{\bar{I}(\lambda)} = K_{0} + \bigl[ K_{1} \cos(\phi/2) + K_{2} \sin(\phi/2)\bigr]^{2}, $$
(3)
where \(\bar{I}\) is the average intensity over all possible tuning phases [\(\phi\)] and \(K_{0}\), \(K_{1}\), and \(K_{2}\) characterize the \(I\)-ripple. Although this specific equation was originally derived for \(I\)-ripple resulting from a misalignment-like feature in the Lyot half-wave plate, it proved to be equally good for other sources of \(I\)-ripple, such as a combination of a tilt in the entrance polarizer and quarter-wave plate of a Michelson interferometer. Therefore this same equation is used to model the \(I\)-ripple of the two Michelson interferometers.
Figure 11 shows the temporal evolution of the measured \(I\)-ripple (in terms of peak-to-peak amplitude) for each of the three tunable elements since the beginning of the mission. It is obtained by fitting the \(I\)-ripple of each tunable element to the intensities of a detune sequence using Equation (3). In a detune sequence, the wavelength tuning of the filter elements are varied independently, allowing one to distinguish features in the individual elements. The peak-to-peak variation in the transmitted intensity [\(I\)] is computed as \(K_{1}^{2}+K_{2}^{2}\).
Unfortunately, it is not possible to determine whether the result of each fit includes only \(I\)-ripple effects, or if other time-dependent imperfections in the tunable elements – those that are not included in our transmittance model – also bleed into these results. Regardless, Figure 11 shows that the intensity transmitted by the tunable elements varies with time. Currently, the \(I\)-ripple is not taken into account when computing the observables. Including this effect when fitting the detune sequences does improve the goodness of fit, as might be expected when adding parameters. However, including the \(I\)-ripple in the filter-transmission profile does not significantly improve the issues identified in the observables. In particular, there was no positive impact on the 24-hour oscillations detailed in Section 3.2.
Computation of Line-of-Sight Observables with the MDI-Like Method
This section describes the algorithm implemented to compute the LoS observables. It is called the MDI-like method because it is based on the technique employed to produce the SOHO/MDI observables.
MDI was designed so that the FWHM of its filter-transmission profiles matches the FWHM of the Ni i line, and the four equally spaced wavelength samples cover an interval equal to twice this FWHM. Consequently, with MDI nearly all of the spectral power in the solar-line shape is captured by the first Fourier coefficients and the phase derived from the cosine and sine components is an estimate of the line position.
HMI was not designed this way, because the dynamic range corresponding to twice the FWHM of the Fe i line is too small to accommodate the large velocity variations resulting from the SDO orbit.
The HMI Implementation of the MDI Algorithm
The HMI algorithm has been described elsewhere (e.g., Equations (4) – (14) here are from Couvidat et al., 2012a). That article provides further details about how it is implemented in the LoS HMI-observables pipeline. For each of the \({\approx}\, 12\) million illuminated pixels on an HMI image, the MDI-like algorithm starts by estimating the first and second Fourier coefficients \(a_{n}\) and \(b_{n}\) (with \(n=1\) or \(n=2\)) of the Fe i line profile \(I(\lambda)\), where \(\lambda\) is the wavelength:
$$ a_{1} = \frac{2}{T} \int_{-\frac{T}{2}}^{+\frac{T}{2}} I(\lambda) \cos \biggl(2\pi \frac{\lambda}{T} \biggr) \mathrm{d}\lambda ;\qquad b_{1} = \frac{2}{T} \int_{-\frac{T}{2}}^{+\frac{T}{2}} I(\lambda) \sin \biggl(2\pi \frac{\lambda}{T} \biggr) \mathrm{d}\lambda $$
(4)
and
$$ a_{2} = \frac{2}{T} \int_{-\frac{T}{2}}^{+\frac{T}{2}} I(\lambda) \cos \biggl(4\pi \frac{\lambda}{T} \biggr) \mathrm{d}\lambda ;\qquad b_{2} = \frac{2}{T} \int_{-\frac{T}{2}}^{+\frac{T}{2}} I(\lambda) \sin \biggl(4\pi \frac{\lambda}{T} \biggr) \mathrm{d}\lambda , $$
(5)
where \(T\) is the period of the observation wavelength span. Nominally, \(T=6 \times 68.8=412.8~\mbox{m}\mathring{\mathrm{A}}\), i.e. six times the nominal separation between two HMI filter transmission profiles. Subsequent discretization of Equations (4) and (5), e.g. as in Equation (11), requires the assumption that the Fe i line profile is periodic with period \(T\).
We assume that the Fe i line has the following Gaussian profile:
$$ I(\lambda)=I_{c}-I_{d} \exp \biggl[- \frac{(\lambda-\lambda_{0})^{2}}{\sigma^{2}} \biggr], $$
(6)
where \(I_{c}\) is the continuum intensity, \(I_{d}\) is the line depth, \(\lambda_{0}\) is the Doppler shift, and \(\sigma\) is a measure of the line width [\(\mathrm{FWHM}=2 \sqrt{\log(2)} \sigma\)].
The Doppler velocity \(v=\mathrm{d}v/\mathrm{d}\lambda \times \lambda_{0}\) can be expressed as
$$ v=\frac{\mathrm{d}v}{\mathrm{d}\lambda} \frac{T}{2\pi} \mathrm{atan} \biggl( \frac{b_{1}}{a_{1}} \biggr), $$
(7)
where \(\mathrm{d}v/\mathrm{d}\lambda= 299792458.0~\mbox{m}\,\mbox{s}^{-1}/6173.3433~\mathring{\mathrm{A}}=48562.4~\mbox{m}\,\mbox{s}^{-1}\,\mathring{\mathrm{A}}^{-1}\). The second Fourier coefficients could also be used,
$$ v_{2}=\frac{\mathrm{d}v}{\mathrm{d}\lambda} \frac{T}{4\pi} \mathrm{atan} \biggl(\frac{b_{2}}{a_{2}} \biggr). $$
(8)
The line depth [\(I_{d}\)] estimate is then equal to
$$ I_{d} = \frac{T}{2 \sigma\sqrt{\pi}} \sqrt{a_{1}^{2}+b_{1}^{2}} \exp \biggl(\frac{\pi^{2}\sigma^{2}}{T^{2}} \biggr), $$
(9)
while \(\sigma\) is equal to
$$ \sigma = \frac{T}{\pi\sqrt{6}} \sqrt{\log \biggl( \frac{a_{1}^{2}+b_{1}^{2}}{a_{2}^{2}+b_{2}^{2}} \biggr)}. $$
(10)
However, HMI samples the iron line at only six points, and therefore what we compute is a discrete approximation to the Fourier coefficients and not the actual coefficients. For instance,
$$ a_{1} \approx \frac{2}{6} \sum _{j=0}^{5} I_{j} \cos \biggl(2\pi \frac{2.5-j}{6} \biggr). $$
(11)
The \(b_{n}\) are determined by a similar formula with cosine replaced by sine. In the LoS observables code, these \(a_{n}\) and \(b_{n}\) are calculated separately for the LCP (\(I+V\)) and RCP (\(I-V\)) polarizations. Applying Equation (7) returns two velocities: \(v_{\mathrm{LCP}}\) and \(v_{\mathrm{RCP}}\).
Departing from the assumptions made, the actual Fe i line profile is not Gaussian (e.g. see Figure 20 in Section 3.2). Moreover, the discrete approximations to \(a_{n}\) and \(b_{n}\) are not accurate as a result of the reduced number of sampling points and because the HMI filter-transmission profiles are not \(\delta\)-functions. Consequently, the observables calculated are relative to the Fe i line convolved with the filters. Therefore, \(v_{\mathrm{LCP}}\) and \(v_{\mathrm{RCP}}\) need to be corrected.
This is the role of look-up tables. They are determined from a realistic model of the Fe i line at rest in the quiet Sun and from calibrated HMI filter-transmission profiles. Look-up tables are described in Section 2.9.2. The sensitivity tables vary across the HMI field of view, since each CCD pixel samples a different ray bundle in the filters. The look-up tables are linearly interpolated at \(v_{\mathrm{LCP}}\) and \(v_{\mathrm{RCP}}\) to derive corrected Doppler velocities (\(V_{\mathrm{LCP}}\) and \(V_{\mathrm{RCP}}\)).
Calibration of the HMI filters shows residual errors at the percent level in their transmittances and free spectral ranges (FSRs), resulting in imperfect look-up tables. The SDO orbital velocity is known very accurately and can be used to somewhat improve these tables. In the HMI pipeline this additional step is referred to as the polynomial correction (see Section 2.10). As currently implemented, this process corrects for the slow drift in Michelson phases and FSRs, but leaves a residual variation in observables at the SDO orbital period.
Since the Sun has magnetic fields and the observations are made in LCP and RCP polarizations, the actual velocity computed will be roughly the centroid of the unsplit and one or the other of the Zeeman-split components with the relative strength depending on the direction of the field. It is convenient that the splitting of the centroids is a good measure of the LoS component of the magnetic field.
Finally, the resulting \(V_{\mathrm{LCP}}\) and \(V_{\mathrm{RCP}}\) velocities are combined to produce a Doppler-velocity estimate:
$$ V = \frac{V_{\mathrm{LCP}}+V_{\mathrm{RCP}}}{2} , $$
(12)
while the LoS magnetic flux density [\(B\)] is estimated as
$$ B = (V_{\mathrm{LCP}}-V_{\mathrm{RCP}}) \; K_{m}, $$
(13)
where \(K_{m} = 1.0/(2.0 \times 4.67 \times 10^{-5} \lambda_{0} g_{L} \mathrm{c})=0.231~\mbox{G}\,\mbox{m}^{-1}\,\mbox{s}\), \(g_{L}=2.5\) is the Landé g-factor, and c is the speed of light (see Norton et al., 2006).
An estimate of the continuum intensity [\(I_{\mathrm{c}}\)] is obtained by reconstructing the solar line from the Doppler-shift, line-width, and line-depth estimates:
$$ I_{\mathrm{c}} \approx \frac{1}{6} \sum _{j=0}^{5} \biggl[I_{j} + I_{d} \exp \biggl(-\frac{(\lambda_{j}-\lambda_{0})^{2}}{\sigma^{2}} \biggr) \biggr], $$
(14)
where \(\lambda_{0}\), \(I_{d}\), and \(\sigma\) are values retrieved using Equations (7), (9), and (10), and \(\lambda_{j}\) are the nominal wavelengths corresponding to each filter profile.
The intensity computations are implemented slightly differently in the HMI pipeline. Tests on synthetic Gaussian lines using proper HMI filter transmittances show that the theoretical algorithm overestimates the line width of Gaussian lines by \({\approx}\, 20~\%\) for a line with \(I_{d}=0.62\) and \(\sigma=0.0613~\mathring{\mathrm{A}}\) (values in Norton et al., 2006). Conversely, the line depth is underestimated by \({\approx}\, 33~\%\). The continuum intensity seems only slightly underestimated (by \({\approx}\, 1~\%\)). We surmise that these errors in the parameters of synthetic Gaussian lines arise because the number of wavelength samples is small and the filters are not \(\delta\)-functions. Unlike velocity shifts (and therefore magnetic-field strength), the line width and line depth are not corrected by look-up tables. In the current implementation, the line depth [\(I_{d}\)] and line width [\(\sigma\)] returned by Equations (9) and (10) are multiplied by \(K_{2} = 6/5\) and \(K_{1} = 5/6\), respectively, so that both values are expected to be closer to the actual ones. The integral of a Gaussian is proportional to \(I_{d} \times \sigma\); therefore the continuum intensity remains unchanged.
Finally, when computing \(I_{d}\) and \(I_{c}\) with Equations (9) and (14), we find that the observed \(\sigma\) is sometimes spurious in the presence of a locally strong magnetic field, especially for pixels away from the solar disk center. For this reason, the observed \(\sigma\) derived from Equation (10) is not used to compute \(I_{d}\) and \(I_{c}\). Instead, a nominal line width is determined from a fifth-order polynomial fit to an azimuthal average around the solar disk center of an HMI line-width map obtained during a period of low solar activity using Equation (10) and corrected by \(K_{1}\), as a function of center-to-limb distance.
The line width, line depth, and continuum intensity are computed separately for both LCP and RCP, and the average is saved in the observables data series.
Observables Look-up Tables
The MDI-like algorithm computes a discrete estimate of the first and second Fourier coefficients of the Fe i line profile from the six measured wavelengths of the observables sequence. Under ideal circumstances, the phases of these two Fourier components would vary linearly with velocity, but in reality, this is only an approximation. Therefore, a correction to the phase-derived raw velocities is made. This is the role of the look-up tables.
It is possible to compensate for real conditions by simulating an observation with a model of the solar line convolved with the actual filter-transmission function. The line parameters used for this process are described in Section 2.9.3 and the maps of individual components of the tunable filter are described in Section 2.8. The simulated intensity signals are computed for the six tuning positions of the filters, the Fourier coefficients are calculated, and the velocity is determined as an expected instrumental response to each input sample velocity. Since the filters vary slowly across the field, the sensitivity table is computed on a \(128 \times 128\) grid that covers the active portion of the CCD for a range of 821 input velocities in steps of \(24~\mbox{m}\,\mbox{s}^{-1}\); thus the maximum input velocity range is \({\pm}\, 9840~\mbox{m}\,\mbox{s}^{-1}\). This range accommodates the signal contributions due to the \({\approx}\, 3500~\mbox{m}\,\mbox{s}^{-1}\) SDO orbit, \({\approx}\,2000~\mbox{m}\,\mbox{s}^{-1}\) solar rotation, \({\approx}\,400~\mbox{m}\,\mbox{s}^{-1}\) supergranulation, and \({\approx}\, 1000~\mbox{m}\,\mbox{s}^{-1}\) from granulation and \(p\)-mode signals (at HMI resolution). This leaves at least \({\approx}\, 3400~\mbox{m}\,\mbox{s}^{-1}\) to account for Zeeman splitting, which introduces an equivalent \(2.16~\mbox{m}\,\mbox{s}^{-1}\,\mbox{G}^{-1}\) splitting from the zero-field line center (see Equation (13)).
The resulting look-up tables have dimensions \(1642 \times 128 \times 128\), where the first dimension is the number of test velocities times two, because the look-up tables store both the first and second Fourier velocities, and the second and third dimensions are the \(x\) and \(y\) grid locations. The tables are stored in several different DRMS series: hmi.lookup, hmi.lookup_corrected, hmi.lookup_expanded, and hmi.lookup_corrected_expanded where “corrected” refers to the fact that the phase maps have been corrected for the CalMode interference fringe pattern (Section 2.8.2), and “expanded” refers to look-up tables computed for a larger off-limb radius.
When the correction is made, a 1642-element sensitivity table of output velocity values is interpolated for each CCD pixel from the look-up tables saved on the coarser \(128 \times 128\) grid. The inverse of the sensitivity table is used to interpolate the input solar velocity as a function of the raw output velocity determined by the MDI algorithm. As the input velocity shifts the line away from the center of the tuning range, the sensitivity decreases.
Figure 12 shows example sensitivity tables for the pixel at CCD center for the front camera for the three different HMI line-profile calibrations over the course of the mission (Section 2.9.3). Tables are computed for each camera. At disk center the front- and side-camera sensitivities differ by at most \(4~\mbox{m}\,\mbox{s}^{-1}\) over the entire velocity range. Each time that HMI is retuned to compensate for the wavelength drift in the Michelson interferometers, new sensitivity tables are produced.
Line Profile Tweaking and Calibration Changes
To produce the look-up tables for the MDI-like algorithm as described in Section 2.9.2, we need a model of the Fe i line profile (at rest and in the quiet Sun). Figure 13 shows, in black, two observed Fe i line profiles: one from the NSO atlas (Kurucz et al., 1984) derived from the Fourier Transform Spectrometer at the McMath–Pierce Solar Facility at Kitt Peak and the other from the solar telescopes at Mount Wilson Observatory (R. Ulrich, private communication, 2011). The two observed line profiles are rather different in terms of depth, width, and asymmetry. Differences arise because instruments have different point spread functions (PSFs) and quite different wavelength resolutions, not to mention targeting, spatial resolution, and other effects. For this reason it is difficult to determine a correct line profile, and consequently, we take a somewhat empirical approach. The detune sequences regularly taken by the HMI and used to estimate the filter and Fe i profiles are sensitive to a combination of different quantities. The fitting code used to estimate the best parameters for the filters and the solar lines also has to deal with partly degenerate quantities (for instance, the filter contrast and the Fe i line depth). Finally, some of the quantities characterizing the HMI filters can only be measured in a laboratory because they require access to a large wavelength range and are not known with a high precision. For instance, the filter-element free spectral ranges (FSR) were measured several times prior to the SDO launch and yielded inconsistent results. For all of these reasons, it is not possible to precisely measure the Fe i line profile using detune sequences. To accommodate the wavelength drift in the Michelson interferometers and other long-term changes in HMI, we have had to retune the instrument several times and to change our estimate of the Fe i line profile to produce better look-up tables. Three different calibrations have been used since the SDO launch in which the solar line profile used to produce the look-up tables has been slightly modified. Figure 13 shows, in orange, the three line profiles used so far.
In the code producing the look-up tables for the MDI-like algorithm, the Fe i line is approximated by a Voigt profile (Tepper-García, 2006) and two Gaussians (to simulate the line asymmetry):
$$\begin{aligned} I = & I_{g}- d_{g} \exp\bigl(-l^{2}\bigr) \biggl( 1-\frac{a}{\sqrt{\pi} l^{2}} \biggl[\bigl(4 l^{2}+3\bigr) \bigl(l^{2}+1\bigr) \exp\bigl(-l^{2}\bigr)-\frac{2 l^{2}+3}{l^{2}} \mathrm{sinh}\bigl(l^{2}\bigr) \biggr] \biggr) \\ &{}- A \exp\bigl(-(\lambda+B)^{2}/C^{2}\bigr) \\ &{}+ D \exp\bigl(-(\lambda-E)^{2}/F^{2}\bigr) , \end{aligned}$$
(15)
where \(l=\lambda/w_{g}\) and for \(|l| \le 26.5\).
Coefficients for the three calibration intervals are given in Table 4. These are sometimes referred to as Calibration Versions 11, 12, and 13. The first calibration was used from 1 May 2010 until 18 January 2012 at 18:15 UT, the second from 18 January 2012 at 18:15 UT to 15 January 2014 at 19:18 UT, and the third has been used ever since. In each case, we used the following FSRs for the filter elements: \(168.9~\mbox{m}\mathring{\mathrm{A}}\) for the NB Michelson, \(336.85~\mbox{m}\mathring{\mathrm{A}}\) for the WB Michelson, \(695~\mbox{m}\mathring{\mathrm{A}}\) for the Lyot element E1, \(1417~\mbox{m}\mathring{\mathrm{A}}\) for E2, \(2779~\mbox{m}\mathring{\mathrm{A}}\) for E3, \(5682~\mbox{m}\mathring{\mathrm{A}}\) for E4, and \(11{,}354~\mbox{m}\mathring{\mathrm{A}}\) for E5.
Table 4 HMI calibration coefficients by time interval.
Polynomial Velocity Correction
After calibrating the HMI Doppler signal as described above, it was clear that an artifact proportional to the SDO orbital velocity remained. A procedure was developed to use the accurately known spacecraft velocity to make a post-calibration correction to the \(V_{\mathrm{LCP}}\) and \(V_{\mathrm{RCP}}\) used to compute the Doppler velocity and LoS magnetic field. This was done by computing a third-order polynomial fit to a 24-hour segment of the difference between the uncorrected full-disk median Doppler velocity rawmedn and the known Sun–SDO radial velocity obs_vr as a function of rawmedn,
$$ \mbox{\textsc{rawmedn - obs\_vr}} = C_{0} + C_{1} \mbox{\textsc{rawmedn}} + C_{2} \mbox{\textsc{rawmedn}}^{2} + C_{3} \mbox{\textsc{rawmedn}}^{3}. $$
(16)
The expectation was that this would account for the slow phase drift of the Michelsons, a possible error in the narrow-band Michelson FSR, and a nonlinear contribution that is probably due to errors in the look-up tables. It had been noticed that the median, which would be expected to be nearly a constant offset from obs_vr, varied through the day in a slowly changing systematic way.
Figure 14 shows the difference between rawmedn and obs_vr plotted as a function of rawmedn for four days or intervals at different seasons throughout the mission. On one of the days, 12 October 2010, a spacecraft-roll maneuver took place (upper left panel). See Section 3.1 for a discussion of the effects of instrument roll. Otherwise, the daily velocity difference generally follows a smooth curve that can be well fit using Equation (16). The residual variability, which includes solar signals, is lower than \(5~\mbox{m}\,\mbox{s}^{-1}\) for a given rawmedn; however, the daily variation in the difference is 15 – \(60~\mbox{m}\,\mbox{s}^{-1}\). The overall offset from zero drifts with time and is determined by the tuning of the instrument filters. We note that the range in rawmedn and the range in the velocity difference are not the same for each day and depend on the orbit of the spacecraft and of Earth. The correction determined by the fit is applied to the measured \(V_{\mathrm{LCP}}\) and \(V_{\mathrm{RCP}}\) velocity at each pixel in the image, and the corrected median velocity is given in the keyword datamedn.
Figure 15 shows the temporal variation of the four polynomial coefficients of the fit since 1 May 2010. \(C_{0}\) represents the offset between rawmedn and obs_vr, and it increases with time primarily because of the wavelength drift in the two Michelson interferometers. The discontinuous jumps in \(C_{0}\) are due to the retuning of the instrument. Jumps in the other coefficients reflect those and other changes to the calibration procedures. Coefficient \(C_{1}\) (upper right) shows sharp discontinuities when the calibration changes (see Table 5), suggesting a temporal dependence of the instrument FSR that is decreasing with time. The polynomial coefficients are computed for times separated by 12 hours and recorded in the DRMS data series hmi.coefficients. The analysis is run only on the NRT data in hmi.V_45s_nrt and not on the definitive data. We note that because the NRT data are not permanently archived, reproducing the fits exactly is impractical.
Table 5 HMI tuning and recalibration times.
The correction is applied to the \(V_{\mathrm{LCP}}\) and \(V_{\mathrm{RCP}}\) returned by the MDI-like method by first interpolating the nearest sets of daily coefficients before and after the t_obs of each computation and then computing a correction for each circular polarization, e.g.
$$ v_{\mathrm{LCP}} = V_{\mathrm{LCP}} - \bigl( C_{0}^{\prime}+ C_{1}^{\prime}V_{\mathrm{LCP}} + C_{2}^{\prime}V_{\mathrm{LCP}}^{2} + C_{3}^{\prime}V_{\mathrm{LCP}}^{3} \bigr), $$
(17)
where \(C_{n}^{\prime}\) are the interpolated polynomial coefficients. The resulting \(v_{\mathrm{LCP}}\) and \(v_{\mathrm{RCP}}\) are used to compute the corrected Doppler velocity and LoS magnetic-field products.
The polynomial correction suffers from three main shortcomings. First, it is computed for the daily velocity range covered by obs_vr, which is quite limited compared to the full velocity range of the solar signal. Consequently, for velocities outside the range of obs_vr, the polynomial correction amounts to an extrapolation, and this can be problematic for a cubic fit. Some of the 12- and 24-hour variations seen in strong LoS magnetic fields may be due to the side effects of this extrapolation. Second, the correction is computed for a spatial average over the CCD, and therefore local differences are not taken into account. In particular, the correction seems to be less accurate toward the solar limb. Finally, the correction coefficients are computed every 12 hours to capture the slow change in shape with time of the plots in Figure 14. However, because the likely errors in the FSRs and the phase maps are changing very slowly in time, the present procedure probably introduces some unnecessary noise in the LoS Doppler and magnetic-field products. The HMI team continues to explore better calibration procedures.
This correction, which may in part correct for an error in the FSR, is not applied to the Stokes observable or down-stream vector magnetic-field inversion. This is consistent with the presence of a 24-hour systematic variation seen even in weak fields in the vector-field products, which we discuss in Section 3.2.
Daily Variation in LoS Doppler-Velocity Data
Even after applying the polynomial velocity correction, a residual error remains in HMI Dopplergrams. After removing the three components of the SDO velocity projected onto each pixel, what remains can be described as an offset from zero plus a nearly linear scale error that changes slowly with time of year. The mean value has a 12-hour period with peak-to-peak variation of \({\approx}\,20~\mbox{m}\,\mbox{s}^{-1}\). We calculate a fit to the residual solar velocity pattern using three terms: solar differential rotation using the traditional form \(V_{\mathrm{rot}}(\lambda) = V_{\mathrm{eq}} - B \sin^{2}(\lambda) - C\sin^{4}(\lambda) \) with \(\lambda= \mbox{latitude}\), but with \(B\) set equal to \(C\); a term equivalent to a rotation in the N–S direction; and radially symmetric limb shift (also known as a convective blue shift). The equatorial rotation rate [\(V_{\mathrm{eq}}\)] from these fits has nearly the same daily shape as obs_vr with a typical amplitude of \({\approx}\, 1.5~\%\) of obs_vr. The differential rotation terms [\(B = C\)] often show a mix of 12-hour and 24-hour periods with peak-to-peak variations of \({\approx}\, 40~\mbox{m}\,\mbox{s}^{-1}\) about the average of \({\approx}\, 300~\mbox{m}\,\mbox{s}^{-1}\). An example of the fit equatorial rotation velocity is shown in Figure 16 for 5 January, 2015. On this day, the peak-to-peak variation in the equatorial rotation rate was \({\approx}\, 65~\mbox{m}\,\mbox{s}^{-1}\). The N–S “rotation” term, which should be zero if the roll angle is correct, shows only small variations about a (much too high) average value of \({\approx}\, 60~\mbox{m}\,\mbox{s}^{-1}\). These values are about the same when the fit is restricted to the inner half of the disk area, showing that the issue is not just with values near the limb.
These numbers suggest that the reported velocity has both scale errors and errors across the field that are not corrected and possibly even exaggerated by the polynomial-correction coefficients. If the polynomial correction is not applied, the equatorial rotation error amplitude is significantly reduced, but the mean amplitude is larger.
A scale error, such as a 1 % error in the FSR, will produce a similar signal in the fit rotation rate. An uncorrected FSR error will introduce both 24- and 12-hour periodicity into the vector field inversions. However, it is not that simple. The N–S error suggests an error in the phase map of one or more of the filter elements. The HMI team is continuing to study these issues.