Introduction to adjoint techniques and the MM5 adjoint modeling system

Abstract

- In OI, 3D-VAR and 3D-PSAS the underlying assumption tells that all the observations are available at the time when the estimation is performed, and analyses are performed at exact intervals depending on their update cycle. Background information is propagated one cycle. This does not take into account the temporal evolution. - The a priori model on the evolution of the state vector is supposed to reflect the evolution with a certain degree of uncertainty. It has the form of a sde (discretized spde with random (noise) variable size of the state vector). The model noise W(t) is a 0 mean and white process. - Information available in 4D data assimilation: i) a background term xb with its covariance matrix B (beginning of the assimilation time period), ii) meteorological model, iii) observations which are distributed in time. - Two approaches to solve the 4D problem: 1. filtering solution is sequential (OI and Kalman filter), 2. smoothing solution aims to globally estimate the state on a complete time period. Smoothing provides better results since it processes more information. - In the 4D problem they construct the minimization such that everything including the model noise (as a function of time) is a parameter of the functional. That's the reason why they need to solve an Augmented Lagrangian functional. - One crucial trick to get rid of the derivative of the state wrt time is to use integration by parts for the lagrangian. The important point here is that the model can be nonlinear. - Lagrangian becomes the adjoint state associated to state x, adjoint model is constructed by linearizing the model and transposing its differential form. - The information in the adjoint model is introduced by the forcing terms which express the deviation of the model prediction from the corresponding observations. The adjoin model can be interpreted as a computational operator which propagates backward the gain of information that results from the observations. - If you assume that the model is perfect in the 4D variational formulation (cov(model noise) = 0, E{W} = E{WW'} = 0) then the resulting Euler-Lagrange equations boil down to 4D-Var algorithm. In this case: i) the direct and adjoint equations are not coupled except at the initial time, ii) at any time the state is uniquely defined by the initial state. Therefore, if N is the dimension of the initial state vector, these N degrees of freedom are enough to fully describe the complete model trajectory. As a result the direct and adjoint equations can be integrated separatelybut sequentially. - adjoint model = tangent linear model = differential model - Parameter estimation is similar to initial condition estimation in 4D-Var. You include it in the minimization and solve for the adjoint equation which now includes the effect of the observations on the parameters as well.