The classical technique of stepwise regression provides a paridigm for variable selection in the linear least squares problem. Trust region methods which control the size of the correction to the current solution estimate prove attractive for nonlinear least squares problems because of their good global convergence behaviour. Recently there has been a convergence of these techniques with the realisation that the $l_1$ trust region method also provides a form of variable selection. These results are reviewed here, and computational methods discussed.

This paper focuses on physics and mathematics behind the numerical software. Our calculations show that the temperature variation recorded in the Great Artesian Basin significantly affects thermodynamic properties of groundwater such as dynamic viscosity and density. Dynamic viscosity varies by a factor of four within the temperature difference typical of the basin, and water density variation may produce a convective motion in higher permeability regions. We suggest to treat the flow in the higher permeability regions as a non-isothermal flow. A rigorous approach to modelling such a flow will be to invoke the continuity equation for a variable density fluid, the energy equation and the momentum equation in the form of the generalised Darcy's law. Accordingly, numerical packages for simulating non-isothermal flows in porous media should be applied.

In ensemble predictions, particularly in numerical weather forecasts, the true initial errors are usually not known, and can be optimally represented by an ensemble of perturbations. Thus the growth rate and predictability of an ensemble of perturbations have direct impact on the quality and predictability of an ensemble forecast. In this paper, local metric entropy (LME) is introduced and used as a measure of local error growth of an ensemble of perturbations in chaotic dynamical systems. The predictability time scale of a dynamical system during a given period of time can also be estimated with higher accuracy by using the LME. It is shown that LME, at any time during the evolution of a dynamical system, can be calculated as the sum of all the positive local Lyapunov exponents. LME does not depend upon the amplitudes nor the configurations of initial perturbations, it depends on the positive local LEs which are intrinsic properties of dynamical systems. LME is adopted as a measure of local error growth to examine the dependence of an ensemble of perturbations to the dynamics of the system. In analysing local error growth rates, LME is compared with the local Lyapunov exponents, normal modes, optimal modes which are all commonly used in meteorological applictions. The correlations between LME, locally largest Lyapunov exponent, the first local Lyapunov exponent, growth rates of the first normal mode and first optimal perturbation are studied. When LME is used to estimate the predictability time scale of a system over a specified time period, it is found that the time scale defined by the LME is closer to the autocorrelation times for some variables than the commonly used Lyapunov time and Kolmogorov - Sinai time in the two dynamical systems we have tested.