Inferences on process noise in a linear model

The linear model $y=G\beta +r+ϵ$ is considered, in which $ϵ$ represents measurement noise, and $r$ represents ‘process’ noise. The two noise terms are assumed to be independent of one another, zero mean, with $cov\left(r\right)=u^2{C}_0,cov\left(ϵ\right)=M$ . Here, it is assumed that $\beta ,u^2$ are unknown, but that the matrices $C_0,M$ are known. Interest here focuses on inferences for the parameter $u^2$ . This may be viewed as a slight generalization of the weighted least squares model $y=X\beta +ϵ$ with $E\left(ϵ\right)=0,cov\left(ϵ\right)={\sigma }^2{W}^{‐1}$ where $W$ is a known positive definite weighting matrix, and $\beta ,{\sigma }^2$ are unknown. This model finds applications in estimating delays through the ionosphere experienced by signals sent by navigation satellites.