The adjoint method
The adjoint of the forward model is used to bring the model simulation into consistency with available observations. The method minimizes a quadratic model-data misfit, (also called cost function; Eq. 1), weighted by the prior data uncertainties, by adjusting control variables iteratively.
\[J(C_{ini},C_{atm}(t))=\sum_{t=1}^{T1}[y(t)-E(t)x(t)]^T[R(t)]^{-1}[y(t)-E(t)x(t)]+C_{ini}^TP(0)^{-1}C_{ini}+\overline{C_{atm}}^TQ_m^{-1}\overline{C_{atm}}+\sum_{t=0}^{T1}C_{atm}^'\left(t\right)^TQ_a^{-1}C_{atm}^'(t)\ \ \left(1\right)\]