an estimate of the signal, and, if both bandpowers share a common map, noise variance. In the estimated covariances of off-diagonal blocks for which the instrumental noise term is included, e.g., the ($95\times 150, 150\times 220$) covariance, the uncertainty in the covariance estimate can be large compared to the true covariance. Therefore we calculate the elements of these off-diagonal blocks analytically from the diagonal elements of the on-diagonal blocks using the equations in Appendix A of L10.

### Field weighting {#sec:fieldweighting}

We calculate a weight, $w^{i}$, for each field and frequency cross spectrum in order to combine the individual-field bandpowers. Unlike R12 who use an $\ell$-dependent weighting scheme, we set $w^{i}$ for each frequency combination equal to the average of the inverse of the diagonal of the covariance matrix over the bins $2500<\ell<3500$. Note that this covariance matrix includes noise and sample variance from simulations, but no beam or calibration errors. We expect and observe that the optimal weights vary slowly with angular scale, and these angular scales are chosen to maximize the sensitivity to the SZ signals.

The combined bandpowers are calculated from the individual-field bandpowers according to: $$D_b = \sum_{i}D_{b}^{i}w^{i}$$ and the covariance matrix similarly: $$\textbf{C}_{bb^\prime} = \sum_{i}w^{i}\textbf{C}_{bb^\prime}^{i}w^{i}.$$ 