power estimation may be done using the difference maps.

As a first approximation assume that the noise is white. However the real observations have different accuracy on the sphere. The COBE orbit configuration was so that the points near the ecliptic poles were observed much longer than others. Let us consider the methods to take into account the measurement accuracy is not equal for different points.

Let the noise in every point on the sphere be normal with zero mean and the variance $\sigma^2(\theta,\varphi)=\sigma_0^2/N(\theta,\varphi)$, where $N(\theta,\varphi)$ is a number of measurements in the point with the $\theta$, $\varphi$ coordinates. Then we may propose the following noise spectrum model on the sphere. If there is a white noise sample $G=\sum n_l^m Y_l^m$, $\langle n_l^m
n_{l^
{\prime}}^{m^{\prime}} \rangle = 0$, $\langle (n_l^m)^2
\rangle =n^2=const$, the true noise on the sphere is a product of $G(\theta,\varphi)$ (here, and only here $n$ is a magnitude of the noise power spectrum, not a spectrum index), and $1/\sqrt{N(\theta,\varphi)}$: $$2
 R(\theta,\varphi) = 1/\sqrt{N(\theta,\varphi)} G(\theta,\varphi) =
w(\theta,\varphi)G(\theta,\varphi)$$

According to Peebls (1980), assume $w(\Omega)Y_l^m(\Omega) =
\sum_{l^{ \prime} m^{\prime}} w_{l l^{\prime}}^{m m^{\prime}}
 Y_{l^{\prime}}^{m^{\prime}}(\Omega)$, where $$w_{l l^{\prime}}^{m m^{\prime}} = \int d \Omega w(\Omega)
 Y_{l^{\prime}}^{m^{\prime}}(\Omega) Y_l^m(\Omega) \$$

For the true noise spherical 