We will construct a non-local, temporal systematics model which allows us to avoid the harsh exclusion criterion for preventing self-subtraction, while still capturing the systematic noise. The mathematical idea of using time-series regression models, as explained in Section 4.3, has been used very successfully for transit observation using the Kepler spacecraft (e.g., Wang et al., 2016). This is a form of temporal differential imaging (TDI, Fig. 4.1, top row), in which temporal differences in the signal are inherent to the data (not induced by the method
of observation), and the systematic trends are caused, for example, by pointing drifts, and other instrumental effects, and are shared by multiple pixels11 on the detector.
The inclusion of many regressors (pixels in training-set or derived quantities thereof) make data-driven models very flexible, such that they can explain even complex systematic behavior. At the same time we have to worry about overfitting the signal.
Another important aspect of data-driven modeling is the selection of the data that should be used as input for constructing the model (the regressors). The scientist’s understanding of the underlying causal structure in the data plays an important role in the selection of the training set. In the Kepler spacecraft example, this could mean, for example, selecting stars of similar magnitude to the target star.
In this work we show how a similar approach can be implemented for pupil tracking high-contrast imaging data.
4.4.1 Implementation and application to high-contrast imaging data
The situation is very similar for high-contrast imaging data. Causes of systematics, with the exception of detector artifacts (e.g., bad/warm/dead pixels, flat field), usually influence either the image globally (e.g., most atmospheric effects, Strehl ratio), or a significant region of the detector (e.g., wind-driven halo, (mis-)alignment of the coronagraph). Even slowly evolving changes in the quasi-static speckle pattern caused by the instrument usually are not strictly confined to one region of the detector (e.g., speckle patterns can display symmetries). A detailed study about an objective optimal choice of regressors is beyond the scope of this work, but multiple heuristic choices can be made: 1) preference to pixels at same separation (brightness), 2) inclusion of pixels located on the exact opposite side of the star, 3) pixels at similar position angle. Figure 4.3 shows an example for the reference pixel (regressor) selection geometry chosen in this work, for an assumed companion that passes 20 pixels north of the host star at the midpoint of the observation.
There are many possible choices for the exact implementation of our algorithm. As the main objective of this work is to demonstrate the principle, we try to achieve a balance between simplicity and effectiveness. This means: a) we use a linear model with a quadratic objective function (here, χ2 statistics), b) we use principal components of the regressor pixel
light curves, not the pixels themselves (principal component regression). This reduces co- linearity in the systematics model and transforms it into an orthogonal basis. It also allows us to forgo regularization in favor of truncating the principle components after a certain number, c) we fit each pixel individually instead of all affected pixels at the same time, d) we fit the planet model and the systematics model simultaneously in order to prevent over- fitting, and thus, e) we employ a complete forward modeling approach, i.e. fit the contrast of the planet for any assumed position, similar to ANDROMEDA (Cantalloube et al., 2015) for direct imaging, or Foreman-Mackey et al. (2015) for Kepler. As such, algorithmic throughput corrections as commonly used in direct imaging pipelines to correct for over-fitting, are not necessary. Also, as we do not use the pixels affected by the planet signal, i.e. self-subtraction (a specific form of over-fitting), is avoided. Unless we perform a blind-search in which we
11for simplicity we use the term “pixels” here, but the same can be applied to spaxels, microspectra, or
any more general quantized “measurements”. More generally, one can also use quantities derived from the pixels such as principal components, instead of the pixels themselves, as will be discussed later.
4 A temporal, non-local systematics model for direct detection of exoplanets at very small angular separations
Figure 4.3: Example of reference pixel selection for one assumed planet position. The signal here is assumed to move through the position (∆x, ∆y) = (0, 20) pixel above the star, at the midpoint of the observation sequence. The reference pixels are shown in white. This example is based on the 51 Eri observation’s parallactic angles.
do not a-priory know the model of what we are looking for, or the model of the signal is too complicated to describe analytically, a forward model is always preferable to speckle subtraction. An example where forward models are difficult to implement are protoplanetary disks, that can have multiple rings, spirals and asymmetries, and generally require detailed hydrodynamical simulations to model. On the other hand, most debris disks can be well modeled using analytic descriptions and radiative transfer codes (Olofsson et al., 2016).
The resulting algorithm looks as follows for one assumed relative position (∆ra,∆dec) of a point source on-sky:
1. Generate forward model of signal at assumed position for the time-series, i.e. embed model PSF12 at appropriate parallactic angle for each exposure to obtain the set of
light curves S for all affected pixels. The set of affected pixels for this (on-sky) position will be called Y . Optional: exclude all bad pixels from Y and S.
2. Construct training set for systematic trends T of non-signal pixel with desired con- straints (heuristically chosen here: similar distance, mirrored area on other side of star, strips of pixels at same position angles as signal just inwards and outwards of sig- nal). Recommended: exclude all bad pixels and/or pixels affected by a known signal. 3. Compute temporal noise model from T : singular value decomposition (SVD) of scaled
training pixels, obtain orthogonal base vector matrix B. This matrix contains our systematics light curves or in transit terminology, the temporal trends. We add another constant column, to account for constant offsets in our data. This constitute our regressor matrix for the noise systematics and is the same for all pixels in Y , although the coefficients will vary.
4. For each pixel p in Y , i.e. affected by signal:
a) Add additional column to B containing the light curve shape for the signal (e.g., planet) for this specific pixel from S, we call this complete systematics+signal matrix Be
b) System of equationsB we = dp, where w is the vector of weight coefficients and dp
is the time-series of pixel p.
c) Solve above system using a χ2-fit of Be to p to obtain optimized coefficients:
w= [BeTC−1Be]−1[BeTC−1dp], where C is assumed to be the unity matrix. The
variance associated to the coefficients are then σw = [BeTC−1Be]−1. The last
coefficient of wp and σw correspond to the planet model. d) Alternative to 4c: marginalize over systematics model
5. Remove significant outliers (e.g., remaining bad pixels that could not be fit properly) 6. Perform average of planet coefficient weighted with their uncertainties over all pixel in
S.
Iterating over a grid of possible positions allows us to construct a conditional contrast map, i.e. the contrast and its uncertainty given the position of an assumed object relative to the central star. From these we can construct a signal-to-noise (SNR) map that can be used for detection. However, the uncertainties have been computed under the simplified assumption of independent and Gaussian noise, which does not accurately reflect the reality
12We use an unsaturated PSF obtained without the coronagraph directly before or after the sequence,
but artificially induced satellite spots could be used. Other ways of reconstructing the PSF such as especially designed coronagraphs that act as focal plane wave-front sensors (Wilby et al., 2017) could in principle be used.
4 A temporal, non-local systematics model for direct detection of exoplanets at very small angular separations
Figure 4.4: Comparison of fitted lightcurve systematics model (with and without planet) with data.
and complexity of high-contrast imaging data. Multiple studies have shown residuals after post-processing with high-contrast imaging pipelines, although significantly whitened, to not be strictly Gaussian and independent (e.g. Marois et al., 2008a). The most direct solution would be to account for these effects in the likelihood function used, but in practice this proves to be challenging since the uncertainty distributions depend both on the observing conditions and instrument. Therefore, we apply the same solution to this problem as in ANDROMEDA Cantalloube et al. (2015) and subsequently pyKLIP (Ruffio et al., 2017) and empirically calibrate the SNR map using the azimuthal standard deviation13 of the
SNR map itself as a function of separation to normalize the map. If not specified otherwise, any reference to S/N in this work refers to the calibrated S/N. This calibrated SNR map will also be referred to as the (normalized) detection map. Any known sources in the field should be masked out when deriving the empirical calibration. At the smallest separations (< 3λ/D) small-sample statistics may become important for the empirical normalization aspect (Mawet et al., 2014). As this work focuses on demonstrating the performance of this method in direct comparison to another method, all contrasts shown in this work are without small sample statistics or coronagraphic transmission correction.
13In our case we use the robust standard deviation based on the median absolute deviation for the
detection limits to be less affected by remaining outliers and real signals.