Incremental cost function method
Minimization of the cost function: \(\displaystyle J(x) = { (x- x^b)^2 \over 2\, \sigma_b^2} +{ [ y^o - {\cal G}(x)]^2 \over2 \, \sigma_r^2}\) with \(\displaystyle {\cal G}(x) = {q \over x - h_L}\).
Linearization of \(\cal G\) around \(x^b\): \(\displaystyle {\cal G}(x) = {\cal G}(x^b+ \delta x) = {\cal G}(x^b) + \mathbf G \, \delta x +O[(\delta x)^2]\)
with \(\displaystyle \delta x = x- x^b\) and \(\displaystyle\mathbf G = {\cal G}'(x^b) =- { q\over (x^b - h_L)^2}\).
Incremental cost function \(\displaystyle J(x) = J_{inc}(x^b + \delta x) + O[(\delta x)^2]\) :
\(\displaystyle J_{inc}(x^b + \delta x) = {(\delta x)^2 \over 2\, \sigma_b^2 } +{ ( d - \mathbf G \, \delta x)^2 \over 2 \, \sigma_r^2} \) where \(d = y^o- {\cal G}(x^b)\) is the innovation