Finite Difference of Adjoint or Adjoint of Finite Difference

Abstract Adjoint models are used for atmospheric and oceanic sensitivity studies in order to efficiently evaluate the sensitivity of a cost function (e.g., the temperature or pressure at some target time tf, averaged over some region of interest) with respect to the three-dimensional model initial conditions. The time-dependent sensitivity, that is the sensitivity to initial conditions as function of the initial time ti, may be obtained directly and most efficiently from the adjoint model solution. There are two approaches to formulating an adjoint of a given model. In the first (“finite difference of adjoint”), one derives the continuous adjoint equations from the linearized continuous forward model equations and then formulates the finite-difference implementation of the continuous adjoint equations. In the second (“adjoint of finite difference”), one derives the finite-difference adjoint equations directly from the finite difference of the forward model. It is shown here that the time-dependent sensiti...