Note that the sum of the weights $\sum_{i}w^{i} = 1$.

### Beam and calibration uncertainties {#sec:beamunc}

For the cosmological analysis, we fold beam and calibration uncertainties into the bandpower covariance matrix. This follows the treatment of beam uncertainties laid out by K11 and used by R12. Calibration uncertainty mimics a constant fractional beam uncertainty.

As in K11, we estimate a beam-calibration correlation matrix for each year: $$\pmb{\rho}^{\rm beam}_{bb^\prime} = \left(\frac{\delta D_b}{D_b}\right) \left(\frac{\delta D_{b^\prime}}{D_{b^\prime}}\right)$$ where $$\frac{\delta D_b}{D_b} = 1-\left(1+\frac{\delta B_b}{B_b}\right)^{-2},$$ and $B_b$ is the Fourier transform of the map of the beam averaged across bin $b$.

We then combine the correlation matrix for each year according to the yearly weights, which is the sum of the fraction of the weights from each field in a given year at a given frequency. We add to this the calibration correlation matrix defined by $\pmb{\rho}^{\rm cal} = \sigma_{\rm cal}^2$, where $\sigma_{\rm cal}$ is the fractional calibration uncertainty. The correlation matrix is transformed into a covariance matrix by multiplying by the measured bandpowers:

$$\textbf{C}^{\rm beam+cal}_{bb^\prime} = \pmb{\rho}^{\rm beam+cal}_{bb^\prime}D_{b}D_{b^\prime}.$$

We add this beam and calibration covariance to the bandpower covariance matrix which contains the effects of sample variance, and instrumental noise. The final covariance matrix includes 