Jitter Correction Algorithms for the COROT Satellite Mission

The COROT space telescope is a French‐led European mission with significant international participation: ESA, Austria, Belgium, and Germany contribute to the payload, and Spain and Brazil contribute to the ground segment. COROT will use the CNES Proteus small‐satellite bus and will be placed in a polar orbit by a Soyuz/ST to be launched in 2006 October. The COROT mission has two scientific programs, both requiring long, uninterrupted observations with very high photometric accuracy. They work simultaneously on adjacent regions of the sky. For more information on the scientific goals of the mission and its current status, we refer the reader to the COROT Web page.³ The expected daily download capacity for COROT is 1.4 Gbits, using three ground stations (S band; Boisnard & Auvergne 2004). The asteroseismology channel is sampled every second: 10 stars, 10 background windows, and four offset windows (which serve to estimate the background light levels and the CCD readout offset). The exoplanet channel is summed over 10 minutes (with some selected windows read out every 32 s) but has some 10,000 targets. The amount of data is thus too large to download every pixel (this would be about 10³ Gbits per day). A mask is assigned to each star on the asteroseismology channel for the duration of an observing run in such a way as to maximize the signal‐to‐noise ratio (S/N). The data that are downloaded for each of the 10 stars each second are the sum of the signal seen in the pixels within the mask, in ADUs (Φ), and barycenter position (x, y). A similar data reduction is applied to the exoplanet channel, but we focus here on finding an instantaneous correction factor for the asteroseismology flux (Φ) lost due to pointing jitter, based on the available auxiliary downlink information (i.e., barycenter position).

The work discussed here was done as part of the Belgian and Brazilian contribution to the COROT project and as such was tailored to the specifications and needs of the COROT mission. However, these methods are relevant to any photometric space mission in which jitter will affect the flux detected from objects being studied. There is already a clear example with COROT: the exoplanet channel may also benefit from the results of the angle fitting for the asteroseismology channel, as described in § 3.3. Other future space missions requiring high‐accuracy photometry, such as Kepler (Koch et al. 1998) and Eddington (Roxburgh & Favata 2003), could make use of similar methods to correct for flux inaccuracies. The problem of losing flux due to satellite motion is not new, nor is it unique, and our methods can be more generally applied.

Previous missions have also had to deal with the problem of small satellite motion, but an extensive search has found no papers published on the topic of correcting for this jitter. It may be the case that previous work has only been the subject of internal documents, making this paper all the more relevant. Preliminary results of this study were presented in the Small Satellite session at the 2005 IAF conference in Japan (see Drummond et al. 2005)

COROT has set a goal of extremely high precision photometry. We need to be able to detect flux variations at a level of 4 ppm (parts per million). It is therefore very important that the flux does not vary significantly for reasons other than real changes in the stars under observation. A movement of the star image on the CCD surface changes the integrated flux because the pixels do not all have the same response (pixel response nonuniformity; PRNU), and also because the flux passes outside the mask defined for flux integration. The first factor has been shown to be a small problem (less than 5% of the photon noise; Boisnard & Auvergne 2004), and the second factor is the subject of this study.

The attitude and orbit control system (AOCS) of the Proteus bus normally guarantees 005 (3 σ) pointing stability on each axis (CNES internal documents). This has been improved to 05 rms for COROT, to comply with the stringent noise regime, which requires that the coupled photometry/jitter noise be 10 times lower than the photon noise for a star of magnitude 6 on the asteroseismology channel. This means that COROT will be using two of the observed target stars for ecartometry—their studied motion will be used in the AOCS loop (Boisnard & Auvergne 2004).

The barycenter of the measured flux above a threshold of 400 e⁻ pixel⁻¹ and within a 40 × 40 pixel area is calculated as

The pixels in the mask are very similar to those in the barycenter calculation, but it is better not to simply calculate the barycenter within the mask area; the point‐spread function (PSF) is not uniform, and if a significant area of the PSF falls outside the mask, the barycenter calculation will be erroneous. It is therefore important to reanalyze which pixels are valid every second. The mask remains fixed for the duration of an observation run, unless new masks are uplinked to the satellite.

Methods for barycenter calculation that are more robust against photon noise and CCD pixel variations have been developed (e.g., Anderson & King 2000), but the limited onboard processing power and memory do not allow the implementation of these methods on the spacecraft. As pointed out above, the limits on the downlink bandwidth also prohibit downlinking of the full frames, which would allow more accurate processing on the ground; only the results of the onboard barycenter calculation can be downloaded.

The amount of stellar flux that falls outside the mask during an exposure of 0.8 s (+0.2 s readout time) can never be known exactly, but we get information on the mean position during the exposure from the barycenter calculation. The difference in position with respect to the ideal position allows us to judge approximately what percentage of flux is lost. We have devised several methods to calculate the flux lost, each differing in complexity. These are all explained below. Over the course of the work, the methods evolved into the one that we propose to apply for COROT, and we present here the quality of this correction in detail.

COROT light curves are simulated using the COROT instrument model (C. Catala, M. Auvergne, & P. Baron 2002, private communication). This consists of the modeled PSF at each position (six possibilities per CCD), with stellar oscillations and activity added, then jitter, background (flat field), diffuse light, dark current, proton impacts, debris impacts, smearing, quantum efficiency (QE) variations, readout electronics noise, and the analog‐to‐digital conversion. Up to this point, the simulation covers 40 × 40 pixels. Next, we apply the mask and sum within it, and then take the barycenter, to give us what COROT will actually download.

In the simulated data, jitter is produced with rms values of about 10^'' in rotation (φ) and 014 in translation (θ and ψ; see Fig. 1). This can be converted into pixel units by a simple calculation. A pixel on the CCD corresponds to 232, so 014 corresponds to 0.06 pixels rms in translation. However, rotation about the line of sight also corresponds to a two‐dimensional translation of the image. At 2000 pixels from the center of rotation, 10^'' translates into 0.1 pixels rms. Therefore, overall motion due to translation and rotation is never more than 1 pixel rms. This modeled jitter consists of periodic and random components. For example, there is a strong orbital period component seen in the Fourier spectrum. Nominal COROT operations place constraints on the random components; i.e., they must never be more significant than 2 ppm in the Fourier spectrum. The periodic elements may be higher than this, since they are easier to identify. The aim of the algorithms is to correct for both the random and periodic components while preserving power in the oscillation frequency peaks and the activity spectra of all types of stars.

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3.1. Using a Model PSF

The first idea of a correction method is to calculate what part of a model PSF profile falls outside the mask for a set of mispointing possibilities within the error budget. An algorithm takes the perfect, modeled PSF for a point source and moves this PSF by a tiny amount, applies the mask (calculated using the same S/N optimization routine as will be used for COROT), and sees how much of the ideal flux is now outside the mask. The amount the PSF is moved could be called an "intrapixel," 1/nth of a pixel. A grid can be built up—a type of "correction surface"—showing how much flux is lost when the barycenter is at n × n intrapixel positions within the central pixel (as stated above, the motion is not expected to be larger than 1 pixel). When Φ, x, and y for a program star are downlinked from COROT, we multiply the flux (Φ) by a correction interpolated from the nearest values to the exact barycenter position (x, y). This correction, referred to as method 1, has the advantage that it can be generated in advance—once one has the true PSF of the instrument optics and the magnitude of the studied object to calculate its mask. An example of the correction surface is shown in Figure 2. It is not symmetric, as neither the PSF nor the mask is symmetric. Here the map divides 1 pixel into 100 × 100 intrapixels. The position where the surface is at 1.00 is the optimal position for the barycenter, where no flux correction is needed. It is clear that knowledge of the PSF is very important when one is considering intrapixel displacement. When conducting tests on the ground, we have access to perfect, noiseless, modeled PSFs. In orbit, at the beginning of each observation run, COROT will download "imagettes" of each star that will be observed in order to check its position and mask allocation and to determine its PSF. About 1000 images will be used to generate an oversampled PSF (detailed to 1/4 pixel). Work is under way to increase the accuracy of these methods.

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3.2. Constructing the Flux‐Loss Map from Measured Data

The second correction method, referred to as method 2, consists of using the data themselves—the flux and barycenter position—to create the correction surface. The barycenter (Δx, Δy) displacement is assigned to a square within the intrapixel grid. Over time, the barycenter falls in the same square more than once. The total flux within the mask for the many exposures when the barycenter is in this square is averaged and compared to the maximum total flux, giving a percentage loss at each position. This obviously needs a long time sample in order to cover the entire possible area with good averaging. Jitter follows a Gaussian distribution, so barycenter positions at large offsets from the center are less well sampled, and the average flux at these positions is thus less well averaged. It follows that the data cannot be corrected immediately. Since orbital periods affect the jitter (with larger motion occurring over the timescale of an orbit), we suggest that at least 10 orbits are needed to have enough information. The time that is needed increases with the precision of the grid (smaller intrapixels mean more time needed to get a good averaging at a position).

Another option is to not correct the extreme points. There are not many of these, so leaving them does not drastically reduce the S/N. This was considered in the algorithms and remains a user option—to apply a threshold of at least 10 values before the correction is considered good and is kept in the correction surface. Figure 3 shows the correction surface for method 2. Instead of covering the full grid, it covers only places where the barycenter position has been detected in simulated COROT data. It is obviously also less smooth. Compared with Figure 2, the surface is also seen to have higher values, meaning that the correction will be less.

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In this method, the accuracy of the barycenter calculation is fundamental. If we apply the wrong correction to a position due to incorrect intrapixel assignment, then the accuracy of the detected flux is not improved.

3.3. Explicit Spacecraft Attitude Reconstruction

It has been mentioned that the accuracy of the barycenter used to judge offset and apply a correction is very important to method 2. This calculation can be affected by photon noise, the PRNU of the CCD, changes in dark current or proton impacts, nonuniform background levels, and many other potential noise sources—it does not just represent the motion of the satellite. The flux will also be affected by these factors. Both of the methods explained so far treat each image on the asteroseismology channel independently. However, they are all on the same spacecraft, and thus the motion due to jitter is the same for all of them, although rotation will affect them differently, depending on their distance from the axis of rotation. A more accurate measure of the displacement per exposure can therefore be found by using the data from all 10 stars. This requires a calculation back to the three Euler angles of the spacecraft in order to take into account the rotational differences.

The spacecraft angles are denoted ψ, θ, and φ, as shown in Figure 1:

where r is the distance from the center of rotation to the ideal center of the star, α is the angle this makes with the x‐axis (see Fig. 4), and s is the size of each pixel, in arcseconds (232 for COROT). Thus, r, α, and hence the displacement change for each star, while the three angles must be the same for the exposure. We employ a least‐squares fit to minimize the difference between the measured values of Δx and Δy and those given by three angles in order to get the best fit for all 10 stars. The fitted angles are used to redetermine the exact barycenter displacement of each star, and these are then used in both of the methods described above in §§ 3.1 and 3.2.

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Once the angles are fitted, we define method 3(i) as method 1 recalculated with the angles, and method 3(ii) as one that uses the new barycenters with the original flux data to generate a new correction surface; these are graphically depicted in Figure 5. Here we see how much smoother the correction surface is once the angles have been fitted. It makes more sense that there is a smooth drop in flux as more of the PSF slides out of the mask. The rough surface in the first correction was due to some barycenter error that the angle fitting has corrected.

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The quality of correction can be judged on the flux itself, or on its power (or Fourier) spectrum. The COROT specifications are mainly based on S/Ns and power at certain frequencies. To complement this, we defined various criteria to judge the quality of each correction method:

• 1.  
  The random jitter noise around 0.01 to 0.1 Hz should be removed. Peaks at the orbital frequency and its harmonics should also be corrected, or at least clearly identified as jitter frequencies and not oscillations. No random frequency peaks of more than 2 ppm should be seen in the power spectrum.
• 2.  
  The oscillation frequencies of the stars should be preserved, even if they are very close to the jitter frequencies. If their power is altered, it should not be by more than the noise correction levels.

The S/N is an important factor in judging the quality of correction, but it can be calculated in many ways, depending on what we consider as signal and what we assume is noise. In the case of a star with many oscillation frequencies, the signal is spread over the power frequency spectrum and is difficult to disentangle from the jitter noise that exists at orbital frequencies and harmonics thereof, as well as at much higher frequencies (0.01 to 0.1 Hz). In order to make results clear for the testing of the jitter correction algorithms, the following points are emphasized:

• 1.  
  The S/N in the flux‐time domain.—This is a reflection of how well the detected or corrected signal represents the input signal (pure oscillations or activity around a mean flux level). This is calculated as

  

  where flux can be the original COROT signal or the corrected version. This means that a brighter star has a higher signal and that a smaller deviation between the flux and the input gives a lower noise value and increases the S/N. By using the standard deviation, we assume a Gaussian distribution to the noise on the flux.
• 2.  
  The power at the known oscillation frequencies.—Since the data are simulated, it is known which oscillation frequencies have been put into the model, so the power in the frequency spectrum at these frequencies can be checked before and after correction. A ratio of the powers at these frequencies before and after correction allows the user to check that none, or very little, of the signal is lost:

  
• 3.  
  The total noise between frequencies 0.01 and 0.1 Hz—This area of the frequency spectrum is almost totally jitter noise. One could look at the mean or the standard deviation of these values, but these in themselves vary according to the brightness of the star. To be consistent for all stars, we have given the result of the integral of the power between 0.01 and 0.1 Hz, over the total input power (integrated over all frequencies). As such, we see how much of the initial measured COROT power is in this frequency band, and how much power of the corrected spectrum is here:

  

In order to give statistical weight to the analysis of the correction quality, many simulations were run. Stars of various effective temperatures and different oscillation frequencies were generated with the same input jitter angles. Data can be generated for six modeled PSFs on each CCD (total 12 stars). These 12 positions have been assigned letters A–L in order to make it easier to identify them (Fig. 6). The first simulations used a simple three‐frequency model: periods of 7200, 900, and 300 s are chosen to represent possible stellar frequencies near the orbital frequency peaks (Table 1). Later on, "real" asteroseismological data were generated to augment the reality of the simulations, using 34 files with 23 to 30 frequencies and their amplitudes selected at random from observed amplitude distributions of different kinds of pulsating stars (De Cat 2002; De Ridder et al. 2004; Breger et al. 2005). Masks were generated as they will be on COROT, and 200,000 s of Φ, x, and y values were saved per star. When the analyses of the corrections are run, different combinations of stars can be chosen to see which factors have an impact on the quality of correction. For example, the angle fitting improves if more stars are chosen at well‐spaced positions on the CCD. If one runs the analysis with only four stars (which is possible, to speed up the process), then the angle fitting is obviously not as accurate as with 10 stars.

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Table 2 shows the improvement in fit. The numbers listed are the standard deviation of the difference between the saved input angles and the fitted angles (arcseconds). The standard deviation evidently gets smaller as the number of stars used increases and their positions from the center of rotation increase. Jitter noise is not the only noise that can be applied in our software. The user can choose which sources of noise to apply, but even the basic simulations contain readout noise and a conversion from electrons to digital units (ADUs). Once the jitter correction seemed to work well on these simple runs, more noise sources were added that could perhaps confuse the algorithms. The largest noise source is photon noise, which does create problems for the correction algorithms.

The flux‐time S/N results we obtained with methods 1, 2, and 3(ii) are shown in Tables 3–pasp_118_844_874tb45. Method 3(i) (using the angles and the modeled PSFs) is not included, because it does not greatly increase the accuracy. As was stated in the description of the method, the major factor in making corrections using a modeled PSF is the accuracy of the PSF, not of the barycenter. The runs were carried out with 10 stars to fit the angles. Only four star results are given: one with activity only, one with oscillations only, one with activity and oscillations, and one modeled as invariable. There is a significant improvement in S/N after correction for the three‐frequency simulation, although this is less significant once noise sources other than jitter are included. The algorithms being tested cannot correct for photon noise, background light, or QE variations. The noise due to jitter is small compared to the noise levels from other sources. The S/N improvement for the 23 or 30 frequency model is still good for method 1, but methods 2 and 3(ii) have suffered. They only manage to improve the star without oscillations. This is only one criterion, however. Improvements are seen in the power ratios and the total noise between 0.01 and 0.1 Hz, as described below.

The power ratios before and after correction are shown in Figures 7–10. Figures 7 and 10 show ratios for stars at positions A and K superposed on the same plot. Methods 1, 2, and 3(ii) give comparable results for the simulation without noise sources other than jitter: the ratios are nearly 1 and are far less spread out that the original COROT signal. Figure 10 shows the power ratios for the same stars as those in Figure 7, but in a simulation with more noise sources. The noise has spread the COROT‐detected power in the oscillation peaks far more widely over the plot, but the correction brings it back closer to 1. As was seen in Table 4, method 1 copes better with more oscillations, and also in terms of power ratios (see Figs. 8 and 9). The asterisks representing this correction are clearly far closer to 1 overall.

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The integrated power in the noise between 0.01 and 0.1 Hz is shown as a fraction (ppm) of the total integrated power in Tables 6–pasp_118_844_874tb78. This again shows that method 1 performs well, but correction 3(ii) is often of a similar quality. In some cases it performs better, and in others slightly worse. It is not obvious from the S/N, but a clear demonstration that correction 3(ii) is sometimes better than correction 1 is given in Figures 11 and 12. Figure 11 shows the power frequency spectrum before and after corrections 1 and 3(ii) for three input oscillation frequencies. Before correction, the COROT data have a complex spectrum. It contains peaks at the orbital frequency (1 × 10⁻⁴ Hz), but also at harmonics of this frequency. There is also a noise addition to the spectrum above 1 × 10⁻² Hz, due to the smaller, faster jitter motion. Correction 1 preserves the input power spectrum but also leaves some of the orbital frequency peaks. Correction 3(ii) has slightly lower power in the true oscillation peaks (as is seen in the power ratio plots) but has completely removed the significant orbital harmonics, while leaving oscillations clearly visible. The orbital period jitter peaks are completely removed, as is the noisy electronics section. Figure 12 shows the results for a star with activity. Again, correction 1 removes some of the jitter peaks, but not all of them, whereas correction 3(ii) removes all of the harmonics and leaves the activity spectrum intact. It is important to remove the orbital frequency and its harmonics, as otherwise it could be misinterpreted as a stellar oscillation frequency. Figure 13 has been included to more clearly represent the flux before and after correction using method 3(ii). The corrected flux clearly contains fast oscillations that cannot be seen clearly in the original flux curve, which is dominated by orbital periods.

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Two methods, and variations thereon, have been devised to correct for flux loss due to small satellite motion. The method using a modeled PSF can give excellent correction, provided that the PSF is known to a high accuracy when creating the correction surface. It should be possible to estimate the PSF to 1/4 pixel accuracy using the 1000 imagettes downloaded at the beginning of an observation run. This method is less sensitive to stellar fluctuations (due to oscillations or activity) over time and requires less time to create. However, if the PSF knowledge is not satisfactory, method 1 can leave significant peaks in the Fourier spectrum due to inaccuracies in the correction surface. The method that uses COROT photometric data to create the correction surface has been shown to work well, assuming that the barycenter accuracy remains on the order of 5 × 10⁻³ pixels. This method requires at least 10 orbits of data to provide a good correction but is independent of the knowledge of the PSF and corrects very well for the orbital frequencies, while preserving the power in the oscillation frequencies.

It has been shown that it can be of benefit to use all of the images to explicitly reconstruct the satellite angular motion and to apply this knowledge to the barycenter displacement. The methods have been tested on stars with varying stellar oscillations, activity, and noise sources. The fitting of the angles works well and benefits both the exoplanet and asteroseismology channels of COROT. It does not always improve the correction method, certainly not by an order of magnitude. This demonstrates the importance of the accuracy of the barycenter calculation, and more simulations should be performed to see if the method can be improved. The methods correct for most of the noise between 0.01 and 0.1 Hz. The levels before and after correction are within the COROT specification limits. The power in the oscillation frequencies is generally preserved. Most importantly, jitter peaks are suppressed while oscillation frequencies right next to them are kept. This is very reassuring for asteroseismology. The user can look at the spectrum and decide whether the correction is necessary, since it does not always need to be applied. The tool that has been developed to compare results from the correction methods is useful to analyze the quality of the COROT signal. It runs fairly quickly on a Sun UltraSPARC 3 750 MHz machine (maximum 1 hr for 10 stars of 200,000 s, with a 200 × 200 intrapixel grid, including fitting the angles).

In this paper we described different methods to correct for jitter noise for high‐precision photometric space missions. A detailed evaluation of the methodology was made by means of simulated data. More statistics are evidently needed on which noise sources confuse the correction once the mission flies. Photon noise is obviously a very large factor and is a significant problem. Jitter can be corrected even when photon noise is present, but the proportion of the noise due to the jitter in this case may be so low that the S/N is not seen to improve greatly. This must be checked on in‐orbit data. For faint stars, in which the mask is smaller and a more significant fraction of the flux is lost to jitter, jitter needs to be corrected, but for bright stars, photon noise is dominant, so the correction may not be needed. While the evaluation of our methodology as presented here is based on simulated data, a forthcoming study will be made in the near future, using laboratory measurements with the COROT CCDs, in addition to in‐orbit MOST data (Walker et al. 2003).