Introduction to Data Assimilation for Scientists and Engineers

Let's assume that \(x\) and \(y\) are scalars and \({\cal G}(x) = G x\) is linear. Let's denote \(B\) and \(R\) the background and observation errors respectively. Show that \(x^a = x^b + K d\) where \(d=y^o-Gx\) (called innovation) and \(K\) a number to be expressed as a function of \(B\), \(R\) and \(G\).