This is valid as well for the adjoint model, which requires additionally a very sophisticated data handling system. The adjoint and tangent linear model are required by 4DVAR and Kalman filtering respectively. However, both are of great importance for sensitivity studies. The variational methods are minimizing the cost function with gradient based algorithms. This algorithms require, of course, the gradient of the cost function regarding the control variables .
Methods used for optimization are usually 'Limited Memory Quasi Newton' algorithms.
As important product this methods are providing an approximation of the Hessian matrix which is the inverse of the covariance matrix of the analysis errors. Different to 4DVAR model errors are treated explicit with their covarianz matrix . Modeling the temporal development of the matrix The Kalman filter is calculating explicitly the temporal development of the forecast error covariance matrix , as it is required by a continuous or cyclic data assimilation. This is really expensive, because integrations of the tangent linear model are required for the integration of . Because of this reason only approximations are used in real, large scale applications.