Solving for the Generalized Inverse of the Lorenz Model (gtSpecial IssueltData Assimilation in Meteology and Oceanography: Theory and Practice)

Abstract

Advanced data assimilation becomes extremely complicated and challenging when used with strongly nonlinear models. Several previous works have reported various problems when applying existing popular data assimilation techniques with strongly nonlinear dynamics. Common for these techniques is that they can all be considered as extensions to methods which have proven to work well with linear dynamics. This paper shows that a weak constraint variational formulation for the Lorenz model, where the full model state in space and time is considered as control variables, can be minimized using a gradient descent method. It is further shown that the weak constraint formulation removes some of the previous reported problems associated to the predictability limit of nonlinear models when strong constraint formulations are used. Further, by using a gradient descent method, problems associated to the use of an approximate tangent linear model when solving the Euler-Lagrange equations or when the extended Kalman filter is used, are eliminated, since the solution is found without integration of any dynamical equations. The method works well with reasonable data coverage and quality of the measurements, however, with poorer data coverage a statistical minimization method, simulated annealing, may be used to search for the global minimum.