Inferences on process noise in a linear model

The linear model y=Gβ+r+ϵy=Gβ+r+ϵ is considered, in which ϵϵ represents measurement noise, and rr represents “process” noise. The two noise terms are assumed to be independent of one another, zero mean, with cov(r)=u2C0,cov(ϵ)=Mcov(r)=u2C0,cov(ϵ)=M. Here, it is assumed that β,u2β,u2 are unknown, but that the matrices C0,MC0,M are known. Interest here focuses on inferences for the parameter u2u2. This may be viewed as a slight generalization of the weighted least squares model y=Xβ+ϵy=Xβ+ϵ with E(ϵ)=0,cov(ϵ)=σ2W−1E(ϵ)=0,cov(ϵ)=σ2W−1 where WW is a known positive definite weighting matrix, and β,σ2β,σ2 are unknown. This model finds applications in estimating delays through the ionosphere experienced by signals sent by navigation satellites.