Appendix A

The purpose of this appendix is to show that using a proper matrix $\tens{C}_{\tens{B}}$, the noise covariance matrix of Eq. (37) given by:  

$$\tens{B'}^{(l,l')}=\tens{C}_{\tens{B}}^{-1} \tens{B}^{(l,l')} {\tens{C}_{\tens{B}}^{\rm T}}^{-1}$$ (A1)

can have a diagonal form. The matrix $\tens{B}^{(l,l')}$ can be diagonalized and we can write.  

$$\tens{B}^{(l,l')}=\tens{P}^{-1} \tens{b}^{(l,l')} {\tens{P}^{\rm T}}^{-1}$$ (A2)

where $\tens{b}^{(l,l')}$ is diagonal and $\tens{P}$ is an orthogonal matrix ($\tens{P}^{-1}=\tens{P}^{\rm T}$). Replacing Eq. (A2) into Eq. (A1), we have:  

$$\tens{B'}^{(l,l')}=\tens{C}_{\tens{B}}^{-1} \tens{P}^{-1} \t... ...')} {\tens{P}^{\rm T}}^{-1} {\tens{C}_{\tens{B}}^{\rm T}}^{-1}.$$ (A3)

Since $\tens{B}^{(l,l')}$ is positive definite all its eigenvalues are positive, therefore the square root of $\tens{b}^{(l,l')}$ is defined. Therefore if we apply the following transformation to the data:  

$$\tens{C}_{\tens{B}}=\tens{P}^{-1} \sqrt{\tens{b}^{(l,l')}}$$ (A4)

we can rewrite Eq. (A3) as:  

$$\tens{B'}^{(l,l')}=\tens{I}$$ (A5)

where $\tens{I}$ is the identity matrix. So replacing $\tens{C}$ in Eq. (33) by $\tens{C}_{\tens{B}}$ will have the effect of removing the artificial correlation due to the noise, and also of performing a normalization. We should point out that the transformation matrix $\tens{C}_{\tens{B}}$ that can achieve this is not unique, and any multiplication by an orthogonal matrix will achieve this. Nevertheless, we give a solution to the problem which can be solved as an eigenvalue and eigenvector problem. The transformation given above does not remove the artificial correlation due the p modes but more or less preserve it. This can have some useful application when one wants to produce spectra with uncorrelated noise but with correlated p-mode signals.