5 Simulations and test of the algorithm

5.1 Methods

We now discuss how we test the method using numerical simulations. For each simulated mission, we produce several maps. The first, which we use as the standard "reference'' of comparison, is a projected map of a mission with only white noise, ${f_{\rm {knee}}=0}$. The remaining maps include 1/f plus white noise streams with ${f_{\rm {knee}}=\eta f_{\rm {spin}}}$, ${\eta \in \{1,2,5,10\}}$. The first of these is an "untreated'' image which is projected with no attempt to remove striping affects. In the second "zero-averaged'' map, we attempt a crude destriping by subtracting its average to each circle. The final "destriped'' map is constructed using the algorithm in this paper. We subtract the input maps (I, Q and U) from the final maps in order to get maps of noise residuals. Note that in case of a zero signal sky, setting the average of each circle to zero is better than destriping by nature because the offsets are only due to the noise. With a real sky, both signal and noise contribute to the average so that zeroing circle not only removes the noise but also the signal. Giard et al. giard99 have attempted to refine their method by fitting templates of the dipole and of the galaxy before subtracting a baseline from each circle. They concluded that an additional destriping (they used the algorithm of Delabrouille  1998) is needed.

5.2 Simulated missions

In order to test its efficiency, the destriping algorithm has been applied to raw data streams generated from simulated observations using various circular scanning strategies representative of a satellite mission as PLANCK, different "Optimized Configurations'', and various noise parameters. The resulting maps were then compared with input and untreated maps to test the quality of the destriping.

The input temperature (I) maps are the sum of galaxy, dipole, and a randomly generated standard CDM anisotropy map (we used HEALPIX and CMBfast). Similarly, the polarization maps Q and U are the sum of the galaxy and CMB polarizations. The CMB polarization maps are randomly generated assuming a standard CDM scenario. For the galactic polarization maps, we constructed a random, continuous and correlated vector field defined on the 2-sphere with a correlation length of $5^\circ$and a maximum polarization rate of ${20\%}$ ($100\%$ gave similar results). Given the temperature map of the sky (not including CMB contribution), we can thus construct two polarization maps for Q and U.

5.3 Results

We first consider the case of destriping pure white noise and check that the destriping algorithm does not introduce spurious structure. Once this is verified, we apply the destriping algorithm to low frequency noise. We find that the quality of the destriping is significantly dependent on $\eta$ only. To demonstrate visually the quality of the destriping, we produce projected sky maps with the input galaxy, dipole and CMB signal subtracted.

For temperature maps, we can compare Figs. 7, 8 and 9. The eye is not able to see any differences between the white noise map and the noise residual on the destriped map. We will see in the following how to quantify the presence of structures. For the "zero-averaged circles'' map, the level of the structure is very high and make it impossible to compute the power spectrum of the CMB (see Fig. 13).

For the Q Stokes parameter, Fig. 10 shows the white noise map, Fig. 11 shows the destriped map for $\eta =1$. Figure 12 shows a map where the offsets are calculated as the average of each circle. The maps for U are very similar. As for the I maps, the destriped map is very similar to the white noise map. There exist some residual structure on the "zero-averaged circles'' map. To assess quantitatively the efficiency of the destriping algorithm, we have first studied the power spectra $C_\ell^T$, ${C_\ell ^E}$, $C_\ell^B$, $C_\ell^{TE}$ and $C_\ell^{TB}$calculated from the I, Q and U maps [Zaldarriaga & Seljak1997,Kamionkowski et al.1997].

Figure 7: The Mollweide projection of the residuals of the I-Stokes parameter for a white noise mission. The scale is in Kelvins. The parameters of the simulation leading to this map are described in Appendix B

Figure 8: The Mollweide projection of the residuals of the I-Stokes parameter after zeroing the average of the circles, for 1/f noise plus white noise with $\eta =1$

Figure 9: The Mollweide projection of the residuals of the Q-Stokes parameter for a white noise mission

Figure 10: The Mollweide projection of the residuals of the ${Q-\rm {Stokes}}$ parameter after zeroing the average of the circles, for 1/f noise plus white noise with $\eta =1$. Although the remaining structures seem small, they are responsible for the excess of power in ${C_\ell ^E}$, see Fig. 15

Figure 11: The Mollweide projection of the residuals of the ${Q-\rm {Stokes}}$ parameter after destriping of 1/f noise plus white noise with $\eta =1$

Figure 12: The Mollweide projection of the residuals of the ${Q-\rm {Stokes}}$ parameter after zeroing the average of the circles, for 1/f noise plus white noise with $\eta =1$. Although the remaining structures seem small, they are responsible for the excess of power in ${C_\ell ^E}$, see Fig. 15

The reference sensitivity of our simulated mission is evaluated by computing the average spectra of 1000 maps of reprojected mission white noise. This reference sensitivity falls, within sample variance, between the two dotted lines represented in Figs. 13, 14, 15 and 16 and below the dotted line in Figs. 17 and 18. Figures 13 and 14 show the spectra $C_\ell^T$ corresponding to ${f_{\rm {knee}}/f_{\rm {spin}}=1\rm {~and~}5}$ respectively, for the T field. Similarly, Figs. 15 and 16 are the spectra ${C_\ell ^E}$. The B field is not represented because it is very similar to E. Figures 17 and 18 represent the correlation between E and T:  $C_\ell^{ET}$. For ${f_{\rm knee}/f_{\rm spin}=1}$, we see that we are able to remove very efficiently low frequency drifts in the noise stream: the destriped spectra obtained are compatible with the white spectrum (within sample variance). Similar quality destriping is achieved for any superposition of 1/f, 1/f² and white noise, provided that the knee frequency is lower than or equal to the spin frequency. In the case of ${f_{\rm knee}/f_{\rm spin}=5}$, the method as implemented here leaves some striping noise on the maps at low values of $\ell$. Modeling the noise as an offset is no longer adequate and a better model of the averaged low-frequency noise is required (superposition of sine and cosine functions for instance), or a more sophisticated method for constructing one circle from 60 scans. We again note that the value of the ratio ${f_{\rm {knee}}/f_{\rm {spin}}}$ for both PLANCK HFI and LFI is likely to be very close to unity (in Fig. 1, ${f^{\rm {measured}}_{\rm {knee}}\sim 0.014}$ Hz and ${f_{\rm {spin}}=0.016}$ Hz).

To quantify the presence of stripes in the maps of residuals, we can compute the value of the "striping'' estimator ${\rm {rms} \left (a^{T,E,B}_{\ell \ell }\right )/\rm {rms}\left (a^{T,E,B}_{\ell 0}\right )}$, because stripes tend to appear as structure grossly parallel to the iso-longitude circles. In the case of pure white noise with a uniform sky coverage, this value is 1. Here, because of the scanning strategy, the sky coverage is not uniform and the value of this estimator is greater than 1, showing that it is not specific of the striping. In order to get rid of the effect of non-uniform sky coverage, we express the estimator ${\rm {rms} \left (a^{T,E,B}_{\ell \ell }\right )/\rm {rms}\left (a^{T,E,B}_{\ell 0}\right )}$in units of ${\rm {rms} \left (a^{T,E,B}_{\ell \ell }\right )/\rm {rms}\left (a^{T,E,B}_{\ell 0}\right )}$ for the white noise. This new estimator is specific to the striping. The results in Table 1 show the improvement achieved by the destriping algorithm although the result is still not perfect.

 

 

Table 1: Values of ${\rm {rms} \left (a^{T,E,B}_{\ell \ell }\right )/\rm {rms}\left (a^{T,E,B}_{\ell 0}\right )}$ as a function of ${\eta ={f_{\rm {knee}}/f_{\rm {spin}}}}$ in units of ${\rm {rms}\left (a^{T,E,B}_{\ell \ell }(\rm {WN})\right )/\rm {rms}}$ ${\left (a^{T,E,B}_{\ell 0}(\rm {WN})\right )}$ We have checked that the systematic difference between the zero-averaged E and B fields is randomly in favor of E and B depending on the particular sky simulation

Method	${f_{\rm {knee}}/f_{\rm {spin}}}$	T	E	B
white noise	0	1	1	1
destriped	0.5	1.19	1.05	1.03
zero-averaged	0.5	51.9	9.23	3.64
undestriped	0.5	6.98	7.04	15.2
destriped	1	1.24	1.12	1.19
zero-averaged	1	52.2	9.91	3.95
undestriped	1	10.7	10.9	7.51
destriped	2	1.26	1.32	1.23
zero-averaged	2	49.5	10.2	3.85
undestriped	2	6.41	9.97	8.18
destriped	5	1.35	1.39	1.38
zero-averaged	5	49.8	10.4	3.99
undestriped	5	11.3	8.24	12.4

Figure 13: Efficiency of destriping for the T field with ${f_{\rm knee}/f_{\rm spin}=1}$. The sample variance associated to a pure white noise mission is plotted as the dotted lines. The "destriped spectrum'' is very close to the white noise spectrum (within the limits due to the sample variance). The zero-averaged and the "not destriped'' spectra are a couple of orders of magnitude above. The solid line represents a standard CDM temperature spectrum and the dashed line represents a CDM temperature spectrum with reionization

Figure 14: Efficiency of destriping for the T field with ${f_{\rm knee}/f_{\rm spin}=5}$. The modelling of low-frequency noise with an offset is no longer sufficient and the destriping leaves some power at low values of $\ell$. Nevertheless, it remains a very good way to significantly reduce the effect of low-frequency noise

Figure 15: Efficiency of destriping for the E field for ${f_{\rm knee}/f_{\rm spin}=1}$. The zero-averaged spectrum is not as bad as for T but the residual striping we can see in Fig. 12 leads to some excess of power for low values of $\ell$ (up to $\ell \sim 100$). We do not see such effect in the destriped spectrum (and maps). The spectra for the B fields are very similar. The solid line represents a standard CDM E spectrum and the dashed line represents a CDM E spectrum with reionization

Figure 16: Same as Fig. 15 but for ${f_{\rm knee}/f_{\rm spin}=5}$. The spectra for the B fields are very similar

Figure 17: Same as Fig. 15 for the ET-correlation for ${f_{\rm knee}/f_{\rm spin}=1}$

Figure 18: Same as Fig. 17 but for ${f_{\rm knee}/f_{\rm spin}=5}$