4 The algorithm

4.1 The general case

To extract the offsets from the measurements, we use a $\chi^2$minimization. This $\chi^2$ relates the measurements $\boldsymbol{M}_{i,j,\delta}$to the offsets O_i and the Stokes parameters $\boldsymbol{S}_{i,j,\delta}$, using the redundancy condition (3). In order to take into account the two contributions of the noise (see Sect. 2) and of the Stokes parameters (see Eq. 4), we model the measurement as:

$$\!\!\!\!\!\!\!\!\! \boldsymbol{M}_{i,j,\delta}= \boldsymbo... ...{S}_{i,j,\delta} \, + \boldsymbol{O}_i \, + \rm {white noise}.$$ (8)

so that we write

$$\begin{multline}\chi^2 = \sum_{i,j\in {\mathcal{I}(i)},\delta=\pm 1} \,\left(\bo... ...\boldsymbol{R}_{i,j,\delta}\, \boldsymbol{S}_{i,j,\delta}\right). \end{multline}$$ (9)

where N_i is the ${h \times h}$ matrix of noise correlation between the h polarimeters.

Minimization with respect to O_i and $\boldsymbol{S}_{i,j,\delta}$ yields the following equations:

$$\begin{gather}\!\!\!\!\!\!\!\!\! {\boldsymbol{N}_i}^{-1} \sum_{j\in \mathcal{I}(... ...ldsymbol{R}_{j,i,-\delta}\, \boldsymbol{S}_{i,j,\delta}\right) = 0. \end{gather}$$ (10) (11)

We can work with a reduced set of transformed measurements and offsets which can be viewed as the Stokes parameters in the focal reference frame and the associated offsets which are the 3 dimensional vectors:

$$\begin{gather}\!\!\!\!\!\!\!\!\! \mathscr{S}_{i,j,\delta} = {\boldsymbol{X}_i}^... ...}^T\, {\boldsymbol{N}_i}^{-1}\, \boldsymbol{\mathcal{A}}. \nonumber \end{gather}$$ (12)

Equations (10) and (11) then simplify to:

$$\begin{gather}\!\!\!\!\!\!\!\!\! \sum_{j\in \mathcal{I}(i),\,\delta=\pm 1} \left... ...dsymbol{R}_{j,i,-\delta}\, \boldsymbol{S}_{i,j,\delta}\right) = 0. \end{gather}$$ (13) (14)

$\boldsymbol{R}_{i,j,\delta}\,\boldsymbol{S}_{i,j,\delta}$ in Eq. (14) can be solved for and the result inserted in Eq. (13). After a few algebraic manipulations, one gets the following linear system for the offsets $\boldsymbol{\Delta}_i$ as functions of the data $\mathscr{S}_{i,j,\delta}$:

$$\begin{multline}\sum_{j\in \mathcal{I}(i),\delta=\pm 1}\left[{\mathchoice {\rm 1... ...boldsymbol{R}}(i,j,\delta)\,\mathscr{S}_{j,i,-\delta}\right] = 0, \end{multline}$$ (15)

where the rotation

$$\!\!\!\!\!\!\!\!\!\widetilde{\boldsymbol{R}}(i,j,\delta)=\boldsymbol{R}_{i,j,\delta} \,{\boldsymbol{R}^{-1}_{j,i,-\delta}}$$ (16)

brings the focal reference frame from its position along scan j at intersection ${\{i,j,\delta\}}$ to its position along scan i at the same intersection (remember that $\{i,j,\delta\} = \{j,i,-\delta\}$). Note that $\widetilde{\boldsymbol{R}}(i,j,\delta)=\widetilde{\boldsymbol{R}}(j,i,-\delta)^{-1}$. In this linear system, we need to know the measurements of the polarimeters at the points ${\{i,j,\delta\}}$ and ${\{j,i, -\delta\}}$. These two points on circles i and j respectively will unlikely correspond to a sample along these circles. So we have linearly interpolated the value of the intersection points from the values measured at sampled points. For a fixed circle i, this is a $3 \times N_{\mathcal{I}(i)}$ linear system. In Eq. (15), i runs from 1 to n, therefore the total matrix to be inverted has dimension $3n \times 3n$. However, because the rotation matrices are in fact two dimensional (see Eq. 7), the intensity components $\Delta_i^I$ of the offsets only enter Eq. (15) through their differences $\Delta_i^I - \Delta_j^I$ so that the linear system is not invertible: the rank of the system is 3n-1. In order to compute the offsets, we can fix the intensity offset on one particular scanning circle or add the additionnal constraint that the length of the solution vector is minimized.

Once the offsets $\boldsymbol{\Delta}_i$ are known, the Stokes parameters in the global reference frame $\left( \hat{e}_{\lambda},\hat{e}_{\beta} \right)$ at a generic sampling k of the circle i, labeled by $\{i,k\}$ are estimated as

$$\!\!\!\!\!\!\!\!\!\boldsymbol{S}_{i,k} = \boldsymbol{R}_{i,k}^{-1} \left(\mathscr{S}_{i,k} - \boldsymbol{\Delta}_i\right),$$ (17)

where R_i,k is the rotation matrix which transforms the focal frame Stokes parameters into those of the global reference frame.

The quantities $\mathscr{S}_{i,k}$ are the Stokes parameters measured in the focal frame of reference at this point and are simply given in terms of the measurements M_i,k (see Eq. 12) by:

$$\!\!\!\!\!\!\!\!\! \mathscr{S}_{i,k} = {\boldsymbol{X}_i}^{-1... ...\mathcal{A}}^T\, {\boldsymbol{N}_i}^{-1} \boldsymbol{M}_{i,k}.$$ (18)

The matrix

$$\!\!\!\!\!\!\!\!\! \boldsymbol{N}^{\rm {Stokes}}_i=\left( \bo... ...boldsymbol{N}_i^{-1} \, \boldsymbol{\mathcal{A}} \right) ^{-1}$$ (19)

is the variance matrix of the Stokes parameters on circle i. Note that this algorithm is totally independent of the pixelization chosen which only enters when reprojecting the Stokes parameters on the sphere.

4.2 Uncorrelated polarimeters, with identical noise

When the polarimeters are uncorrelated with identical noise, the variance matrix reduces to $\boldsymbol{N}_i = {\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}/{\sigma_i}^2$ and the matrices X_i can all be written as

$$\!\!\!\!\!\!\!\!\! \boldsymbol{X}_i = \frac{1}{{\sigma_i}^2}\... ...mbol{X} = \boldsymbol{\mathcal{A}}^T \boldsymbol{\mathcal{A}}.$$

Case of "Optimized Configurations''

We have shown [Couchot et al.1999] that the polarimeters can be arranged in "Optimized Configurations'', where the h polarimeters are separated by angles of $\pi/h$. If the noise level of each of the hpolarimeters is the same and if there are no correlation between detector noise, then the errors of the Stokes parameters are also decorrelated and the matrix X has the simple form:

$$\!\!\!\!\!\!\!\!\! \boldsymbol{X} = \frac{n}{8}\left( \begin{array}{ccc} 2&0&0\\ 0&1&0\\ 0&0&1 \end{array}\right).$$ (20)

Because this matrix commutes with all rotation matrices $\widetilde{\boldsymbol{R}}(i,j,\delta)$, Eq. (15) simplifies further to

$$\begin{multline}\frac{N_{\mathcal{I}(i)}}{\sigma_{\mathcal{I}(i)}^2}\boldsymbol{... ...\boldsymbol{R}}(i,j,\delta) \, \mathscr{S}_{j,i,-\delta}\right), \end{multline}$$ (21)

where the sum over $\delta$ is explicit on the left side of the equation and we have defined an average error $\sigma_{\mathcal{I}(i)}$ along circle i by

$$\!\!\!\!\!\!\!\!\!\frac{N_{\mathcal{I}(i)}}{\sigma_{\mathcal{... ...= \sum_{j\in{\mathcal{I}(i)}} \frac{2}{\sigma_i^2+\sigma_j^2},$$

and where the rotation matrix $\widetilde{\boldsymbol{R}}(i,j,\delta)$ is defined by Eq. (16). Rotations $\widetilde{\boldsymbol{R}}(i,j,\delta)$ and $\widetilde{\boldsymbol{R}}(i,j,-\delta)=\widetilde{\boldsymbol{R}}(j,i,\delta)^{-1}$ correspond to the two intersections between circles i and j. Equation (21) can be simplified further. We can separate the $\boldsymbol{\Delta}_i$ and the $\mathscr{S}_{i,j,\delta}$ into scalar components related to the intensity: $\Delta^I_i,\ \mathscr{S}^I_{i,j,\delta}$ and 2-vectors components related to the polarization: $\boldsymbol{\Delta}^P_i,\ \mathscr{S}^P_{i,j,\delta}$. We obtain then two separate equations, one for the intensity offsets $\Delta^I_i$, which is exactly the same as in the unpolarized case (see Appendix A):

$$\begin{multline} \sum_{j\in {\mathcal{I}(i)}}\frac{2}{\sigma_i^2+\sigma_j^2}\,\... ...mathscr{S}^I_{i,j,\delta} - \mathscr{S}^I_{j,i,-\delta}\right), \end{multline}$$ (22)

and one for the polarization offsets $\boldsymbol{\Delta}^P_i$:

$$\begin{multline}\frac{N_{\mathcal{I}(i)}}{\sigma_{\mathcal{I}(i)}^2}\,\boldsymbo... ...bol{\mathcal{R}}(i,j,\delta)\,\mathscr{S}^P_{j,i,-\delta}\right), \end{multline}$$ (23)

where $\Psi_{ij}=\Psi_1(i,j,\delta)-\Psi_1(j,i,-\delta)$ is the angle of the rotation that brings the focal reference frame along scan j on the focal reference frame along scan i at intersection ${\{i,j,\delta\}}$. The two dimensional matrix $\boldsymbol{\mathcal{R}}(i,j,\delta)$ is the rotation sub-matrix contained in $\widetilde{\boldsymbol{R}}(i,j,\delta)$ (see Eq. 16). Note that all mixing between polarization components have disappeared from the left side of Eq. (23)). Therefore we are left with two different $n \times n$ matrices to invert in order to solve for the offsets $\boldsymbol{\Delta}_i$, instead of one $3n \times 3n$ matrix.

As in Eq. (15), the linear system in Eq. (22) involves differences between the offsets $\Delta_i^I$, the matrix is not invertible, and we find the solution in the same way as in the general case. On the other hand, for the polarized offsets $\Delta_i^P$, the underlying matrix of Eq. (23) is regular as expected.