Approximate Gauss-Newton methods for nonlinear least squares problems

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an “inner” direct or iterative process. In comparison with Newton’s method and its variants, the algorithm is attractive because it does not require the evaluation of second-order derivatives in the Hessian of the objective function. In practice the exact Gauss–Newton method is too expensive to apply operationally in meteorological forecasting, and various approximations are made in order to reduce computational costs and to solve the problems in real time. Here we investigate the effects on the convergence of the Gauss–Newton method of two types of approximation used commonly in data assimilation. First, we examine “truncated” Gauss–Newton methods where the inner linear least squares problem is not solved exactly, and second, we examine “perturbed” Gauss–Newton methods where the true linearized inner problem is approximated by a simplified, or perturbed, linear least squares problem. We give conditions ensuring that the truncated and perturbed Gauss–Newton methods converge and also derive rates of convergence for the iterations. The results are illustrated by a simple numerical example. A practical application to the problem of data assimilation in a typical meteorological system is presented.

Annular variability and eddy-zonal flow interactions in a simplified atmospheric GCM. Part 1: Characterization of high and low frequency behaviour

Experiments have been performed using a simplified, Newtonian forced, global circulation model to investigate how variability of the tropospheric jet can be characterized by examining the combined fluctuations of the two leading modes of annular variability. Eddy forcing of this variability is analyzed in the phase space of the leading modes using the vertically integrated momentum budget. The nature of the annular variability and eddy forcing depends on the time scale. At low frequencies the zonal flow and baroclinic eddies are in quasi equilibrium and anomalies propagate poleward. The eddies are shown primarily to reinforce the anomalous state and are closely balanced by the linear damping, leaving slow evolution as a residual. At high frequencies the flow is strongly evolving and anomalies are initiated on the poleward side of the tropospheric jet and propagate equatorward. The eddies are shown to drive this evolution strongly: eddy location and amplitude reflect the past baroclinicity, while eddy feedback on the zonal flow may be interpreted in terms of wave breaking associated with baroclinic life cycles in lateral shear.