A gradient, ∇xJ, of some scalar measure of a forecast, J, with respect to the vector of model control variables, x, that can include initial conditions, boundary conditions, and parameters. The inner product (δx, ∇xJ), where δx is a perturbation vector and ∇xJ is an adjoint sensitivity vector, provides δJ, a first-order (tangent linear) approximation to the difference, ΔJ, between an unperturbed nonlinear forecast and a nonlinear forecast with control variables perturbed by δx. See adjoint model, tangent linear equation, tangent linear approximation.