Forward Sensitivity Approach to Dynamic Data Assimilation

The least squares fit of observations with known error covariance to a strong-constraint dynamical model has been developed through use of the time evolution of sensitivity functions—the derivatives of model output with respect to the elements of control (initial conditions, boundary conditions, and physical/empirical parameters). Model error is assumed to stem from incorrect specification of the control elements. The optimal corrections to control are found through solution to an inverse problem. Duality between this method and the standard 4D-Var assimilation using adjoint equations has been proved. The paper ends with an illustrative example based on a simplified version of turbulent heat transfer at the sea/air interface. 1. Introduction Sensitivity function analysis has proved valuable as a mean to both build models and to interpret their output in chemical kinetics (Rabitz et al. [1], Seefeld and Stockwell [2]) and air quality modeling (Russell et al. [3]). Yet, the ubiquitous systematic errors that haunt dynamical prediction cannot be fully understood with sensitivity functions alone. We now include an optimization component that leads to an improved fit of model to observations. The methodology is termed forward sensitivity method (FSM)—a method based on least squares fit of model to data, but where algorithmic structure and correction procedure are linked to the sensitivity functions. In essence, corrections to control (the initial conditions, the boundary conditions, and the physical and empirical parameters) are found through solution to an inverse problem. In this paper we derive the governing equations for corrections to control and show their equivalence to equations governing the so-called 4D-Var assimilation method (four-dimensional variational method)—least squares fit of model to observations under constraint (LeDimet and Talagrand [4]). Beyond this equivalence, we demonstrate the value of the FSM as a diagnostic tool that can be used to understand the relationship between sensitivity and correction to control. We begin our investigation by laying down the dynamical framework for the FSM: general form of the governing dynamical model, the type and representation of model error that can identified through the FSM, and the evolution of the sensitivity functions that are central to execution of the FSM. The dual relationship between 4D-Var/adjoint equations is proved. The step-by-step process of assimilating data by FSM is outlined, and we demonstrate its usefulness by application to a simplified air-sea interaction model. 2. Foundation