Appendix A: Destriping of unpolarized data

We give here the formulæ for the simpler case of destriping temperature measurements with bolometers. The assumptions are the same as in the polarized case and we adopt the same notation for the common quantities. Instead of polarimeters, we have h bolometers. Since the measurement is no longer dependent on the orientation of the bolometer, the model of the measurement is given by:

$$\!\!\!\!\!\!\!\!\! \boldsymbol{M}_{i,j,\delta}=\boldsymbol{O}... ...left( \begin{array}{c} 1 \\ \vdots \\ 1 \end{array}\right)$$ (A1)

where $\boldsymbol{M}_{i,j,\delta}$ is the h-vector made of measurements by the h bolometers, O_i is an h-vector containing the offsets for the i'th circle and ${I_{i,j,\delta}}$ is the temperature in the direction of the intersection point labeled by ${\{i,j,\delta\}}$. u is an h-vector, corresponding to the $\boldsymbol{\mathcal{A}}$ matrix of the polarized case. The $\chi^2$ can be written as:

$$\begin{multline}\chi^2 = \sum_{i,j\in {\mathcal{I}(i)},\delta= \pm 1} \,\left(\... ...} - \boldsymbol{O}_i - I_{i,j,\delta} \; \boldsymbol{u} \right). \end{multline}$$ (A2)

In this case, the physical quantity uniquely defined at an intersection point is the temperature of the sky at this point. The constraint used here for removing low-frequency noise is then:

$$\!\!\!\!\!\!\!\!\! I_{i,j,\delta} = I_{j,i,-\delta}.$$ (A3)

Given this relation, the minimization of the $\chi^2$ with respect to O_i and ${I_{i,j,\delta}}$ leads to the linear system:

$$\begin{multline}\sum_{j \in \mathcal{I}(i)} \frac{x_j}{x_i+x_j} \left( \Delta_i... ...eft( \mathscr{I}_{i,j,\delta} - \mathscr{I}_{j,i,-\delta} \right) \end{multline}$$ (A4)

where

$$\!\!\!\!\!\!\!\!\! x_i=\boldsymbol{u^T} \, \boldsymbol{N}_i^{-1} \, \boldsymbol{u}$$ (A5)

corresponds to the X_i matrix of the polarized case,

$$\!\!\!\!\!\!\!\!\! \mathscr{I}_{i,j,\delta}=\frac{1}{x_i} \, ... ...{u^T} \, \boldsymbol{N}_i^{-1} \, \boldsymbol{M}_{i,j,\delta}$$ (A6)

corresponds to the $\mathscr{S}_{i,j,\delta}$ local Stokes parameters of the polarized case and the scalar

$$\!\!\!\!\!\!\!\!\! \Delta_i=\frac{1}{x_i} \, \boldsymbol{u^T} \, \boldsymbol{N}_i^{-1} \,\boldsymbol{O}_i$$ (A7)

corresponds to the 3-vector $\Delta_i$ of the polarized case.

Temperature offsets appear through their differences $\Delta_i - \Delta_j$ so this linear system is not invertible and we can use the same methods to invert the system than in the polarized case. The size of the matrix to invert is $n \times n$.

After evaluation of the offsets $\Delta_i$, one can recover the value of temperature for any sample k along circle i:

$$\!\!\!\!\!\!\!\!\! I_{i,k}=\mathscr{I}_{i,k}-\Delta_i.$$ (A8)