In our recently submitted paper, Tim Brandt and I show how to measure spectral errors and covariances in high-contrast integral-field spectroscopic observations of exoplanets and include them self-consistently in parameter retrievals.
Why is this an interesting/important problem? As we show in our paper, we know some component of the noise in such observations will be spectrally correlated due to the scaling of diffraction speckles with wavelength, and not accounting for this can potentially bias the inferred parameters. As a demonstration of this effect, let's look at some real data:
In the above animation, we step through the wavelength frames of a GPI data cube of the star HIP 21861. There are no other astrophysical sources in this image (e.g., a planet or disk), so we are looking at regions of empty sky, which are realizations of the spectral noise. The colored dots in the left panel correspond to the noise spectra in the right panel, and the vertical black line tracks the current wavelength frame of the data cube. It is immediately clear that the noise is highly correlated – you can see the correlated, radial motion of the speckles in the left panel, and in the right panel, you can even pick out a characteristic correlation length by eye.
For comparison, here's what the animation would look like if the noise were completely uncorrelated:
To generate this movie, I simulated independent Gaussian noise within the footprint of the GPI data cube. As you can see, this familiar white noise static is very different from what we observe. Fortunately, since we often have a plethora of data of empty sky, it is possible to measure both the amplitude and covariance of the spectral noise.
In the figure below, I show the measured correlation function, which is closely related to the covariance matrix, within an annulus (i.e., at fixed separation) extracted from the GPI data cube shown in the first animation (left panel) and the simulated data cube shown in the second animation (right panel). As expected, the correlation function for the simulated data is a delta function, which means the covariance matrix will be diagonal. In contrast, the left panel reveals spectral correlation on several characteristic scales; hence, the covariance matrix will have non-zero off-diagonal terms, which should be included in an atmospheric parameter retrieval.
Interestingly, we find that the form of the covariance matrix is highly sensitive to the details of the implementation of the algorithm used to subtract the point-spread function. This finding highlights that it is essential to measure the spectral covariance for each data set and reduction technique.
For many more details, as well as a study of the impact of including the measured spectral covariances in atmospheric parameter retrievals, check out our paper.