ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: (i) the same ratio between PSF size and pixel width (on the detector) at all wavelengths; (ii) the same incident flux at all wavelengths; (iii) the ratio between diagonal and off-diagonal terms. Some values can be null or negative. While it is highly unlikely that the three conditions will be satisfied exactly, there is always a chance of them almost being satisfied. In which case, instead of null eigenvalues we might see significantly smaller values compared to the others. This results in a poorly conditioned matrix. Having a large range of eigenvalues causes precision problems when trying to invert the matrix. A more careful study might be needed to explore the conditions when negative, null or very small eigenvalues can occur. 4.1.2 Results for the slicer Fig. 2 shows the result of the super-optimal procedure on a slicer simulated supernova. Using SpecSim, we simulated the observation of a Type Ia supernova at redshift 1.5 with a 2-m telescope of focal length 20m with an exposure time of 8h. The simulated errors include the zodiacal light, detector and shot noise. The high-resolution input spectrum, shown in blue, is the spectrum input to the slicer (theoretical Type Ia supernova spectrum scaled to magnitude, mirror area and exposure time). The red error bars display the 1 errors computed by the super-optimal extraction. The choice of the exposure time yields S/N of 10.4 for the Si ii absorption feature at 1.53m. Figure 2. SPECSim simulation of the Type Ia supernova spectrum and extraction with the super-optimal extractor. The input truth (blue) is simulated and reconstructed (red) with 1 error for a supernova at redshift 1.5 observed with the slicer spectrograph on a 2.0-m space telescope of 20.0-m focal length. The exposure time is 8h. The error bars estimated by the super-optimal extractor result from the zodiacal light, detector noise and shot noise. Fig. 3 shows the percentage error to compare the optimal (grey) and super-optimal (red) spectrum extraction methods highlighting the Si ii line region. The input spectrum and the observational parameters used are identical to those used in Fig. 2 . The super-optimal extraction performs significantly better than the optimal extraction. At the redshifted Si ii absorption line at 1.537m, the super-optimal extraction performs 1.6times better in the error percentage than the optimal extraction, and the residuals (deviation from theoretical) are also smaller, with their distribution compatible with the theory of chi-square fit. The goodness of fit can be tested by the pull distribution, which corresponds to the histogram of the ratio of the residuals and estimated error. Ideally, this should be a normal distribution centred at 0 and with rms of 1. This is the case for the super-optimal extraction and not for the optimal extraction. In our case, this is because the latter underestimates the errors. Figure 3. Comparison of the optimal (grey) and super-optimal (red) spectrum extraction