7.2 Correlated Noise
7.2.4 PCA filter
In the Oslo pipeline the three first principal components (the three PCA modes with highest eigenvalues) are removed from the data. We will in this section account for the effects of removing one, and then more, modes from the time stream, as well as analyzing each mode and how they affect the final results. We will refer to the mode corresponding to the highest eigenvalue as the global common mode and other modes as residual common modes.
The major difference between the PCA filter and the other filters is that when applying the other filters, we more or less know how the end product will appear. We set up these filters so that they only remove or alter data in a fashion specified by us. The effects of a PCA filter is hard to anticipate as a principle component analysis decomposes data into modes of varying variance degree. Therefore we wish to start by analyzing the results of removing modes of high variance, until either the remaining data is approximately uncorrelated, the remaining correlations do not change or until we start to alter the signal. We will focus this analysis on detector four and five which we can see from the middle plot in figure 7.15 are widely affected by the remaining correlations.
When studying figure 7.16 we notice that the removal of the global common mode has a significantly larger effect than the removal of residual modes (compared to figure 7.15).
This comes as no surprise as the definition of the first PCA mode is the mode pointing in the direction of most variance. There seems to be relatively strong correlations at the LSB in detector 5. These are the sideband LSB edge effects mentioned in section 6.1.
We notice some effect from removing the second mode as well, most noticeably around the edge frequencies of detector five. We also notice the removal of correlations between detectors as well.
As seen from the bottom plot in figure 7.16 adding the removal of the third principal component has very little effect on the final result. It is however hard to, by the eye, say to which degree each component effects the data. For this reason we add a more quantitative analysis, as seen in figures 7.17 and 7.18. Here we have also included the forth component for comparison.,
From the top bottom plot in figure 7.17 we see how the global common mode ac-counts for close to 60% of the standard deviation for detector four and seven with an average of 9.8%of the standard deviation explained over all detectors. The middle two plots analyze the second mode. The removal of the second principal component affects detectors 15 and 18 the most, and explains 3.6%of the standard deviation of the data.
It is worth mentioning that even though the global common mode should explain more of the overall standard deviation there might still be detectors and channels which are more affected by lower modes as evident when comparing the amplitudes for detector 15 in figure 7.17. The second mode actually has a greater effect on this detector than the first mode, which indicates the importance of analyzing more than one mode. As expected the third component hardly affects the data at all, except for detector six, and the forth component has no significant effect on any of the detectors. Further analysis
Figure 7.16: Correlations after the PCA filter. The top, middle and bottom plots have had the first, first two and first three PCA modes removed respectively.
of the higher order components indicate that the removal of these will not affect the remaining correlation issues before affecting the signal it self, which we do not wish.
By eye we also note that the final time stream at the bottom of figure 7.18 look much more like white noise than does the previous from the same figure.
Figure 7.17: The frequency channel acceptance rate (top plot) and the%of the white noise standard deviation for each detector.
Having performed the analysis above we are confident that the removal of the first three principal components is enough to yield a good white noise estimation. For this reason we now turn our focus to the resulting time stream compared to before the filter was applied as well as the corresponding power spectrum plot. As seen in figure 7.19, it is not very clear from the time stream how the data changes before and after the filter.
In general the amplitude of each oscillation in the time stream seems to decline after
Figure 7.18: The time streams after the removal of each corresponding principal com-ponent.
the filtration, although this is not always the case. The power spectrum from figure 7.19 sheds a little more light on the situation. We clearly see that at low temporal frequencies the power spectrum moves closer to the expected ideal white noise value
than before. At high frequencies the power spectrum stays the same.
Figure 7.19: The data before and after application of the PCA filter. The top figure shows the TOD and the bottom figure shows the binned values of the power spectrum as well as the expected ideal white noise level (dashed line) and the temporal frequency of the scan (dotted line).
In the middle correlation plot we notice what we in section 6.1 referred to as specific separations in the off-diagonal stripes. These correlations are between frequencies with 9-10 channel separation and has so far only been found in detector 6. The correlations can be traced back to the high resolution grip with 1024 channels, although the source of it and why it only shows up in detector 6 is still not known. At the bottom figure we notice the remaining LSB edge effects, which for detector 5 are not subtracted by the removal of the first three principle components.
Figure 7.20: Correlations after the PCA-filter. The middle and bottom plot has a lower color scale, which enables us to see the remaining correlations in more detail on detectors 6-8. The bottom plot is zoomed on feed 8.
7.3 Absolute gain calibration
Let us take a step back and think about what we have done so far. All the efforts of the pipeline so far has been to remove correlated noise from the data in an attempt to establish clean data with white-noise-like properties. The reason we want the data to behave as white noise is that when we average over all the time samples, the white noise will cancel out as it corresponds to random oscillations around mean zero. The signal, being much weaker than the noise, is not supposed to be affected by the PCA filter. Thus when the white noise cancels out we are left with the signal only. How do we know then that the pipeline does not affect the signal? We compare the leftover signal with the signal from a source with a known signal strength. Should our signal strength match the predicted one we can be confident that the pipeline only filter out unwanted noise.
As we calculate the signal we have to take the aperture efficiency (Aeff) into account.
Aperture efficiency is the ratio of power measured by the receiver and the total power from the source incident on the telescope. The expected antenna temperature, Texp as calculated by the pipeline is therefore given by
Texp= Tant Aeff
(7.1) where Tant is the measured antenna temperature from the pipeline and Aeff is the aperture efficiency.
Both Aeff and Texp can be derived from first principle, and we will use these ex-pressions to estimate if our Tant-calculations are accurate. To calculate the expected antenna temperature, which is the physical variable we end up with, of a source we use [54],
Texp =Tjup Ωjup
Ωbeam, (7.2)
whereTJup is the brightness temperature of Jupiter,ΩJup is the solid angle of Jupiter and ΩBeam is the solid angle of the beam. The reason why we are using Jupiter as a calibration source is due to the WMAP experiments [55] extensive brightness temper-ature measurements of it. Since WMAP used other frequency bands than COMAP we perform a weighted power-law fit such that
log10(Tjup) = 0.149 log10 ν
22.8
+ 2.15(K). (7.3)
Similarly we might find the solid angle of the COMAP beam by fitting the peak beam directivity at the frequencies provided by the COMAP beam model.
Ωbeam= (0.2593 ln(ν)2−1.965 ln(ν) + 3.865)10−5(sr) (7.4) The solid angle of the target, in this case Jupiter, can be found by taking the solid angle at the average Jupiter-Earth distance and then scaling it, such that
Ωjup= 2.481×10−8
5.2AU r
2
(7.5)
where r is the Jupiter-Earth distance at the time of measurement.
Aeff if given by
Aeff = TexpGideal Sjup
(7.6) where Gideal is the ideal gain given by 32.5 Jy/K and Sjup is flux of Jupiter. The flux is given by
Sjup = 2kbTjupΩjup
λ2 (7.7)
At channels around 26 GHz the brightness temperature of Jupiter is approximately 144 K. Thus at the channels corresponding to approximately 26 GHz we should measure an antenna temperatures between 1.2 and 2.5 K depending on the Jupiter-Earth distance at the time of the scan. As an example scan we will use scan 4852, which is a circular scan of Jupiter from 6th April 2019, with a Jupiter-Earth distance of 4.915 AU. This should result in an antenna temperature of approximately 1.86 K.
Figure 7.21 shows the measured antenna temperature weighted with the aperture efficiency from eight different detectors as well as the expected antenna temperature.
We see that almost every detector peaks at the expected value at 1.86 K, when we take the noise level into account, which is exactly what we would expect. In addition there are some detectors that peaks well below the expected value. We believe the explanation of this to be that Jupiter is not in the middle of the beam at the time of measurement, thus the full signal is not measured. We see similar results for other detectors and
Figure 7.21: The final antenna temperature for scan 4852 at 26 GHz for the first eight detectors.
frequencies. From this we can assume that the PCA filter does not remove a significant part of the signal, and that my gain estimations seem to be accurate.