A singular vector perspective of 4D-Var: Filtering and interpolation

Four-dimensional variational data assimilation (4D-Var) combines the information from a time sequence of observations with the model dynamics and a background state to produce an analysis. In this paper, a new mathematical insight into the behaviour of 4D-Var is gained from an extension of concepts that are used to assess the qualitative information content of observations in satellite retrievals. It is shown that the 4D-Var analysis increments can be written as a linear combination of the singular vectors of a matrix which is a function of both the observational and the forecast model systems. This formulation is used to consider the filtering and interpolating aspects of 4D-Var using idealized case-studies based on a simple model of baroclinic instability. The results of the 4D-Var case-studies exhibit the reconstruction of the state in unobserved regions as a consequence of the interpolation of observations through time. The results also exhibit the filtering of components with small spatial scales that correspond to noise, and the filtering of structures in unobserved regions. The singular vector perspective gives a very clear view of this filtering and interpolating by the 4D-Var algorithm and shows that the appropriate specification of the a priori statistics is vital to extract the largest possible amount of useful information from the observations. Copyright © 2005 Royal Meteorological Society

Robust eigenstructure assignment in quadratic matrix polynomials: nonsingular case

Feedback design for a second-order control system leads to an eigenstructure assignment problem for a quadratic matrix polynomial. It is desirable that the feedback controller not only assigns specified eigenvalues to the second-order closed loop system but also that the system is robust, or insensitive to perturbations. We derive here new sensitivity measures, or condition numbers, for the eigenvalues of the quadratic matrix polynomial and define a measure of the robustness of the corresponding system. We then show that the robustness of the quadratic inverse eigenvalue problem can be achieved by solving a generalized linear eigenvalue assignment problem subject to structured perturbations. Numerically reliable methods for solving the structured generalized linear problem are developed that take advantage of the special properties of the system in order to minimize the computational work required. In this part of the work we treat the case where the leading coefficient matrix in the quadratic polynomial is nonsingular, which ensures that the polynomial is regular. In a second part, we will examine the case where the open loop matrix polynomial is not necessarily regular.

The new Hollywood, 1981–1999: special/visual effects

The emergence and development of digital imaging technologies and their impact on mainstream filmmaking is perhaps the most familiar special effects narrative associated with the years 1981-1999. This is in part because some of the questions raised by the rise of the digital still concern us now, but also because key milestone films showcasing advancements in digital imaging technologies appear in this period, including Tron (1982) and its computer generated image elements, the digital morphing in The Abyss (1989) and Terminator 2: Judgment Day (1991), computer animation in Jurassic Park (1993) and Toy Story (1995), digital extras in Titanic (1997), and ‘bullet time’ in The Matrix (1999). As a result it is tempting to characterize 1981-1999 as a ‘transitional period’ in which digital imaging processes grow in prominence and technical sophistication, and what we might call ‘analogue’ special effects processes correspondingly become less common. But such a narrative risks eliding the other practices that also shape effects sequences in this period. Indeed, the 1980s and 1990s are striking for the diverse range of effects practices in evidence in both big budget films and lower budget productions, and for the extent to which analogue practices persist independently of or alongside digital effects work in a range of production and genre contexts. The chapter seeks to document and celebrate this diversity and plurality, this sustaining of earlier traditions of effects practice alongside newer processes, this experimentation with materials and technologies old and new in the service of aesthetic aspirations alongside budgetary and technical constraints. The common characterization of the period as a series of rapid transformations in production workflows, practices and technologies will be interrogated in relation to the persistence of certain key figures as Douglas Trumbull, John Dykstra, and James Cameron, but also through a consideration of the contexts for and influences on creative decision-making. Comparative analyses of the processes used to articulate bodies, space and scale in effects sequences drawn from different generic sites of special effects work, including science fiction, fantasy, and horror, will provide a further frame for the chapter’s mapping of the commonalities and specificities, continuities and variations in effects practices across the period. In the process, the chapter seeks to reclaim analogue processes’ contribution both to moments of explicit spectacle, and to diegetic verisimilitude, in the decades most often associated with the digital’s ‘arrival’.