6 Discussions and conclusions

Comparison with other methods.

Although no other method has yet been developped specifically for destriping polarized data, many methods exist for destriping unpolarized CMB data, which could be adapted to polarized data as well.

We first comment on the classical method which consists in modelling the measurement as

$$m_{\rm t} = A_{\rm tp}T_{\rm p} + n_{\rm t}$$ (24)

where A is the so-called "pointing matrix", T a vector of temperatures in pixels of the sky, $m_{\rm t}$ the data and $n_{\rm t}$

the noise. The problem is solved by inversion, yielding an estimator of the signal:

$$\tilde{T_{\rm p}} = [A^{\rm t} N^{-1} A]^{-1} A^{\rm t} N^{-1} \, m,$$ (25)

where $N=\langle nn^{\rm t} \rangle$ is the noise correlation matrix and $A^{\rm t}$ is the transposed matrix of A.

This method can be extended straightforwardly to polarized measurements, at the price of extending by a factor of $3\times h$ the size of the matrix $A_{\rm tp}$, by 3 that of vector $T_{\rm p}$ (replaced by $(I_{\rm p},Q_{\rm p},U_{\rm p})$), and by h that of the data stream (remember that h is the number of polarimeters). The implementation of this formally simple solution may turn into a formidable problem when megapixel maps are to be produced. Numerical methods have been proposed by a variety of authors [Wright1996,Tegmark1997], that use properties of the noise correlation matrix (symmetry, band-diagonality) and of the pointing matrix (sparseness). Such methods, however, rely critically on the assumption that the noise is a Gaussian, stationary random process, which has been a reasonable assumption for CMB mission as COBE where the largest part of the uncertainty comes from detector noise, but is probably not so for sensitive missions such as Planck. Our method requires only inverting a $3n \times 3n$ matrix where n is the number of circles involved, and does not assume anything on the statistical properties of low frequency drifts. It just assumes a limit frequency (the knee frequency) above which the noise can be considered as a white Gaussian random process.

Another interesting method is the one that has been used by Ganga in the analysis of FIRS data [Ganga1994], which is itself adapted from a method developed originally by Cottingham cottingham87. In that method, coefficients for splines fitting the low temperature drifts are obtained by minimising the dispersion of measurements on the pixels of the map. Such a method, very similar in spirit to ours, could be adapted to polarization. Splines are natural candidates to replace our offsets in refined implementations of our algorithm.

Here, we have assumed that the averaged noise can be modeled as circle offsets plus white noise (Eq. 8), i.e. that the noise between different measurements from the same bolometer is uncorrelated after removal of the offset. This allowed us to simplify the $\chi^2$ to that shown in Eq. (9). In reality, the circular offsets do not completely remove the low frequency noise and there does remain some correlation between the measurements. The amount of correlation is directly related to the value of ${f_{\rm {knee}}/f_{\rm {spin}}}$; the smaller ${f_{\rm {knee}}/f_{\rm {spin}}}$ the smaller the remaining correlation. Figures 13-18 already contain the errors induced from the fact that we did not include these correlations in the covariance matrix, and thus demonstrate that the effect is small for ${f_{\rm {knee}}/f_{\rm {spin}}\sim 1}$.

Conclusion.

The destriping as implemented in this paper removes low frequency drifts up to the white noise level provided that ${f_{\rm {knee}}/f_{\rm {spin}}\le 1}$. For larger ${f_{\rm {knee}}}$, the simple offset model for the averaged noise could be replaced with a more accurate higher-order model that destripes to better precision provided the scan strategy allows to do so, as discussed in Delabrouille et al. delabrouille98e. We are currently working on improving our algorithm to account for these effects. However, despite the shortcomings of our model, it still appears to be robust for small ${f_{\rm {knee}}}$ and can serve as a first order analysis tool for real missions. In particular, our technique cannot only be used for the Planck HFI and LFI, but can also be adopted for other CMB missions with circular scanning strategies, such as COSMOSOMAS for instance [Rebolo et al.1999].

Acknowledgements

We would like to acknowledge our referee's very useful suggestions.