Forecast Error Correction using Dynamic Data Assimilation

As neophytes in science, we wondered about predictability. On the one hand, we knew that an eclipse of the sun could be predicted with great accuracy years in advance; yet, some of us wondered why the 2–3 day forecasts of rain or snow were so suspect. Even before we took university courses on the subject, we could speculate, and often incorrectly, as to the reasons that underlie marked difference in predictability of these events.

In this chapter, using a simple deterministic nonlinear, scalar model, we derive the dynamics of evolution of the first-order and second-order forward sensitivities of the solution or model forecast with respect to control (the initial condition and parameters).

Our goal in this chapter is to clarify the intrinsic relation that exists between the so called adjoint sensitivity and the forward sensitivity method (FSM) introduced in Chap. 2 Adjoint sensitivity lies at the core of the well known 4-dimensional variational method known as 4D-VAR (Lewis et al. 2006, Chaps. 22–26).

In this chapter we generalize the results of Chaps. 2 and 3 to nonlinear dynamical systems. In Sect. 4.1, the dynamics of evolution of the first order sensitivities are given. The intrinsic relation between the adjoint and forward sensitivities are explored in Sect. 4.2. In Sect. 4.3 forward sensitivity based data assimilation scheme is derived. Exercises and Notes and References are contained in Sects. 4.4 and 4.5, respectively.

The basic principle of forecast error correction using dynamic data assimilation (DDA) is to alter or move the model trajectory towards a given set of (noisy) observations in such a way that the weighted sum of squared forecast errors (which is also the square of the energy norm of the forecast errors) is a minimum. We have classified the various sources of forecast error in Sect. 1.5.2 A little reflection reveals that there are essentially two ways of altering the solution of a dynamical system: (1) changing the control consisting of the initial/boundary conditions and the parameters of the dynamical model, and (2) by adding an explicit external forcing (a form of state dependent control) which will in turn force the model solution towards the desired goal. The 4D-VAR and FSM based deterministic framework are designed to alter the model trajectory to correct the forecast errors by iteratively adjusting the control (initial/boundary conditions and parameters) using one of the well established algorithms for minimizing the square of the energy norm of the forecast errors. Refer to Chaps. 2 and 4 of this book and Lewis et al. (2006) for details. Identification of errors in the dynamics and associated adjustments go beyond correcting control. Generally, the correction terms added to the constraints force the model forecast to more closely fit the observations either empirically (so-called nudging process) or optimally (a process that involves a least squares fit of model to observation). Lakshmivarahan and Lewis (2013) have comprehensively viewed the work on nudging. In this chapter we address the process of optimally adjusting the constraints to fit the observations. The most powerful method to accomplish this task is Pontryagin’s minimum principle (PMP). It is the centerpiece of this chapter. At the end of the chapter, we also consider an optimal method used in meteorology developed by Derber (1989) and applied to hurricane tracking by DeMaria and Jones (1993).

In the late fall and winter, a rhythmic cycle of cold air penetrations into the Gulf of Mexico (GoM) takes place.

With the steady growth of the interest in ocean circulation systems and their impact on climate change, there has been a predictable increase in the number of tracer/drifter/buoy type ocean observing systems. There is a rich and growing literature on the development and testing of data assimilation technology to effectively utilize this new type of data set. This class of data assimilation has come to be known as Lagrangian data assimilation.