Driving saccade to pursuit using image motion

Within the context of active vision, scant attention has been paid to the execution of motion saccades—rapid re-adjustments of the direction of gaze to attend to moving objects. In this paper we first develop a methodology for, and give real-time demonstrations of, the use of motion detection and segmentation processes to initiate capture saccades towards a moving object. The saccade is driven by both position and velocity of the moving target under the assumption of constant target velocity, using prediction to overcome the delay introduced by visual processing. We next demonstrate the use of a first order approximation to the segmented motion field to compute bounds on the time-to-contact in the presence of looming motion. If the bound falls below a safe limit, a panic saccade is fired, moving the camera away from the approaching object. We then describe the use of image motion to realize smooth pursuit, tracking using velocity information alone, where the camera is moved so as to null a single constant image motion fitted within a central image region. Finally, we glue together capture saccades with smooth pursuit, thus effecting changes in both what is being attended to and how it is being attended to. To couple the different visual activities of waiting, saccading, pursuing and panicking, we use a finite state machine which provides inherent robustness outside of visual processing and provides a means of making repeated exploration. We demonstrate in repeated trials that the transition from saccadic motion to tracking is more likely to succeed using position and velocity control, than when using position alone.

A new frequency-uniform coercive boundary integral equation for acoustic scattering

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L2(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star-shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star-combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second-kind integral operator for which convergence of the Galerkin method in L2(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star-combined operator implies frequency-explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high-frequency case. The proof of coercivity of the star-combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains.

Ensemble prediction for nowcasting with a convection-permitting model - II: forecast error statistics

A 24-member ensemble of 1-h high-resolution forecasts over the Southern United Kingdom is used to study short-range forecast error statistics. The initial conditions are found from perturbations from an ensemble transform Kalman filter. Forecasts from this system are assumed to lie within the bounds of forecast error of an operational forecast system. Although noisy, this system is capable of producing physically reasonable statistics which are analysed and compared to statistics implied from a variational assimilation system. The variances for temperature errors for instance show structures that reflect convective activity. Some variables, notably potential temperature and specific humidity perturbations, have autocorrelation functions that deviate from 3-D isotropy at the convective-scale (horizontal scales less than 10 km). Other variables, notably the velocity potential for horizontal divergence perturbations, maintain 3-D isotropy at all scales. Geostrophic and hydrostatic balances are studied by examining correlations between terms in the divergence and vertical momentum equations respectively. Both balances are found to decay as the horizontal scale decreases. It is estimated that geostrophic balance becomes less important at scales smaller than 75 km, and hydrostatic balance becomes less important at scales smaller than 35 km, although more work is required to validate these findings. The implications of these results for high-resolution data assimilation are discussed.

Rational speculative bubbles: an empirical investigation of the London Stock Exchange

In recent years, a sharp divergence of London Stock Exchange equity prices from dividends has been noted. In this paper, we examine whether this divergence can be explained by reference to the existence of a speculative bubble. Three different empirical methodologies are used: variance bounds tests, bubble specification tests, and cointegration tests based on both ex post and ex ante data. We find that, stock prices diverged significantly from their fundamental values during the late 1990's, and that this divergence has all the characteristics of a bubble.

Conditioning of incremental variational data assimilation, with application to the Met Office system

Implementations of incremental variational data assimilation require the iterative minimization of a series of linear least-squares cost functions. The accuracy and speed with which these linear minimization problems can be solved is determined by the condition number of the Hessian of the problem. In this study, we examine how different components of the assimilation system influence this condition number. Theoretical bounds on the condition number for a single parameter system are presented and used to predict how the condition number is affected by the observation distribution and accuracy and by the specified lengthscales in the background error covariance matrix. The theoretical results are verified in the Met Office variational data assimilation system, using both pseudo-observations and real data.

Conditioning and preconditioning of the variational data assimilation problem

Numerical weather prediction (NWP) centres use numerical models of the atmospheric flow to forecast future weather states from an estimate of the current state. Variational data assimilation (VAR) is used commonly to determine an optimal state estimate that miminizes the errors between observations of the dynamical system and model predictions of the flow. The rate of convergence of the VAR scheme and the sensitivity of the solution to errors in the data are dependent on the condition number of the Hessian of the variational least-squares objective function. The traditional formulation of VAR is ill-conditioned and hence leads to slow convergence and an inaccurate solution. In practice, operational NWP centres precondition the system via a control variable transform to reduce the condition number of the Hessian. In this paper we investigate the conditioning of VAR for a single, periodic, spatially-distributed state variable. We present theoretical bounds on the condition number of the original and preconditioned Hessians and hence demonstrate the improvement produced by the preconditioning. We also investigate theoretically the effect of observation position and error variance on the preconditioned system and show that the problem becomes more ill-conditioned with increasingly dense and accurate observations. Finally, we confirm the theoretical results in an operational setting by giving experimental results from the Met Office variational system.

Advanced global optimisation for mission analysis and design

This report describes the analysis and development of novel tools for the global optimisation of relevant mission design problems. A taxonomy was created for mission design problems, and an empirical analysis of their optimisational complexity performed - it was demonstrated that the use of global optimisation was necessary on most classes and informed the selection of appropriate global algorithms. The selected algorithms were then applied to the di®erent problem classes: Di®erential Evolution was found to be the most e±cient. Considering the speci¯c problem of multiple gravity assist trajectory design, a search space pruning algorithm was developed that displays both polynomial time and space complexity. Empirically, this was shown to typically achieve search space reductions of greater than six orders of magnitude, thus reducing signi¯cantly the complexity of the subsequent optimisation. The algorithm was fully implemented in a software package that allows simple visualisation of high-dimensional search spaces, and e®ective optimisation over the reduced search bounds.

New results on the thermodynamical properties of the climate system

In this paper the authors exploit two equivalent formulations of the average rate of material entropy production in the climate system to propose an approximate splitting between contributions due to vertical and eminently horizontal processes. This approach is based only on 2D radiative fields at the surface and at the top of atmosphere. Using 2D fields at the top of atmosphere alone, lower bounds to the rate of material entropy production and to the intensity of the Lorenz energy cycle are derived. By introducing a measure of the efficiency of the planetary system with respect to horizontal thermodynamic processes, it is possible to gain insight into a previous intuition on the possibility of defining a baroclinic heat engine extracting work from the meridional heat flux. The approximate formula of the material entropy production is verified and used for studying the global thermodynamic properties of climate models (CMs) included in the Program for Climate Model Diagnosis and Intercomparison (PCMDI)/phase 3 of the Coupled Model Intercomparison Project (CMIP3) dataset in preindustrial climate conditions. It is found that about 90% of the material entropy production is due to vertical processes such as convection, whereas the large-scale meridional heat transport contributes to only about 10% of the total. This suggests that the traditional two-box models used for providing a minimal representation of entropy production in planetary systems are not appropriate, whereas a basic—but conceptually correct—description can be framed in terms of a four-box model. The total material entropy production is typically 55 mW m−2 K−1, with discrepancies on the order of 5%, and CMs’ baroclinic efficiencies are clustered around 0.055. The lower bounds on the intensity of the Lorenz energy cycle featured by CMs are found to be around 1.0–1.5 W m−2, which implies that the derived inequality is rather stringent. When looking at the variability and covariability of the considered thermodynamic quantities, the agreement among CMs is worse, suggesting that the description of feedbacks is more uncertain. The contributions to material entropy production from vertical and horizontal processes are positively correlated, so that no compensation mechanism seems in place. Quite consistently among CMs, the variability of the efficiency of the system is a better proxy for variability of the entropy production due to horizontal processes than that of the large-scale heat flux. The possibility of providing constraints on the 3D dynamics of the fluid envelope based only on 2D observations of radiative fluxes seems promising for the observational study of planets and for testing numerical models.

Computing Markowitz efficient frontiers using a spreadsheet optimiser

Markowitz showed that assets can be combined to produce an 'Efficient' portfolio that will give the highest level of portfolio return for any level of portfolio risk, as measured by the variance or standard deviation. These portfolios can then be connected to generate what is termed an 'Efficient Frontier' (EF). In this paper we discuss the calculation of the Efficient Frontier for combinations of assets, again using the spreadsheet Optimiser. To illustrate the derivation of the Efficient Frontier, we use the data from the Investment Property Databank Long Term Index of Investment Returns for the period 1971 to 1993. Many investors might require a certain specific level of holding or a restriction on holdings in at least some of the assets. Such additional constraints may be readily incorporated into the model to generate a constrained EF with upper and/or lower bounds. This can then be compared with the unconstrained EF to see whether the reduction in return is acceptable. To see the effect that these additional constraints may have, we adopt a fairly typical pension fund profile, with no more than 20% of the total held in Property. The paper shows that it is now relatively easy to use the Optimiser available in at least one spreadsheet (EXCEL) to calculate efficient portfolios for various levels of risk and return, both constrained and unconstrained, so as to be able to generate any number of Efficient Frontiers.

Robust pole assignment in linear state feedback

Numerical methods are described for determining robust, or well-conditioned, solutions to the problem of pole assignment by state feedback. The solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized. It is shown that for these solutions, upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized. A measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed-loop poles, and hence the suitability of the given poles for assignment.

Robustness in partial pole placement

The robustness of state feedback solutions to the problem of partial pole placement obtained by a new projection procedure is examined. The projection procedure gives a reduced-order pole assignment problem. It is shown that the sensitivities of the assigned poles in the complete closed-loop system are bounded in terms of the sensitivities of the assigned reduced-order poles, and the sensitivities of the unaltered poles are bounded in terms of the sensitivities of the corresponding open-loop poles. If the assigned poles are well-separated from the unaltered poles, these bounds are expected to be tight. The projection procedure is described in [3], and techniques for finding robust (or insensitive) solutions to the reduced-order problem are given in [1], [2].

Stability results for the time-harmonic Maxwell equations with impedance boundary conditions

We consider the time-harmonic Maxwell equations with constant coefficients in a bounded, uniformly star-shaped polyhedron. We prove wavenumber-explicit norm bounds for weak solutions. This result is pivotal for convergence proofs in numerical analysis and may be a tool in the analysis of electromagnetic boundary integral operators.

Does increased hydrochemical model complexity decrease robustness?

The aim of this study was, within a sensitivity analysis framework, to determine if additional model complexity gives a better capability to model the hydrology and nitrogen dynamics of a small Mediterranean forested catchment or if the additional parameters cause over-fitting. Three nitrogen-models of varying hydrological complexity were considered. For each model, general sensitivity analysis (GSA) and Generalized Likelihood Uncertainty Estimation (GLUE) were applied, each based on 100,000 Monte Carlo simulations. The results highlighted the most complex structure as the most appropriate, providing the best representation of the non-linear patterns observed in the flow and streamwater nitrate concentrations between 1999 and 2002. Its 5% and 95% GLUE bounds, obtained considering a multi-objective approach, provide the narrowest band for streamwater nitrogen, which suggests increased model robustness, though all models exhibit periods of inconsistent good and poor fits between simulated outcomes and observed data. The results confirm the importance of the riparian zone in controlling the short-term (daily) streamwater nitrogen dynamics in this catchment but not the overall flux of nitrogen from the catchment. It was also shown that as the complexity of a hydrological model increases over-parameterisation occurs, but the converse is true for a water quality model where additional process representation leads to additional acceptable model simulations. Water quality data help constrain the hydrological representation in process-based models. Increased complexity was justifiable for modelling river-system hydrochemistry. Increased complexity was justifiable for modelling river-system hydrochemistry.

Validation of ACE-FTS satellite data in the upper troposphere/lower stratosphere (UTLS) using non-coincident measurements

CO, O3, and H2O data in the upper troposphere/lower stratosphere (UTLS) measured by the Atmospheric Chemistry Experiment Fourier Transform Spectrometer(ACE-FTS) on Canada’s SCISAT-1 satellite are validated using aircraft and ozonesonde measurements. In the UTLS, validation of chemical trace gas measurements is a challenging task due to small-scale variability in the tracer fields, strong gradients of the tracers across the tropopause, and scarcity of measurements suitable for validation purposes. Validation based on coincidences therefore suffers from geophysical noise. Two alternative methods for the validation of satellite data are introduced, which avoid the usual need for coincident measurements: tracer-tracer correlations, and vertical tracer profiles relative to tropopause height. Both are increasingly being used for model validation as they strongly suppress geophysical variability and thereby provide an “instantaneous climatology”. This allows comparison of measurements between non-coincident data sets which yields information about the precision and a statistically meaningful error-assessment of the ACE-FTS satellite data in the UTLS. By defining a trade-off factor, we show that the measurement errors can be reduced by including more measurements obtained over a wider longitude range into the comparison, despite the increased geophysical variability. Applying the methods then yields the following upper bounds to the relative differences in the mean found between the ACE-FTS and SPURT aircraft measurements in the upper troposphere (UT) and lower stratosphere (LS), respectively: for CO ±9% and ±12%, for H2O ±30% and ±18%, and for O3 ±25% and ±19%. The relative differences for O3 can be narrowed down by using a larger dataset obtained from ozonesondes, yielding a high bias in the ACEFTS measurements of 18% in the UT and relative differences of ±8% for measurements in the LS. When taking into account the smearing effect of the vertically limited spacing between measurements of the ACE-FTS instrument, the relative differences decrease by 5–15% around the tropopause, suggesting a vertical resolution of the ACE-FTS in the UTLS of around 1 km. The ACE-FTS hence offers unprecedented precision and vertical resolution for a satellite instrument, which will allow a new global perspective on UTLS tracer distributions.

How difficult is it to recover from dangerous levels of global warming?

Climate models provide compelling evidence that if greenhouse gas emissions continue at present rates, then key global temperature thresholds (such as the European Union limit of two degrees of warming since pre-industrial times) are very likely to be crossed in the next few decades. However, there is relatively little attention paid to whether, should a dangerous temperature level be exceeded, it is feasible for the global temperature to then return to safer levels in a usefully short time. We focus on the timescales needed to reduce atmospheric greenhouse gases and associated temperatures back below potentially dangerous thresholds, using a state-of-the-art general circulation model. This analysis is extended with a simple climate model to provide uncertainty bounds. We find that even for very large reductions in emissions, temperature reduction is likely to occur at a low rate. Policy-makers need to consider such very long recovery timescales implicit in the Earth system when formulating future emission pathways that have the potential to 'overshoot' particular atmospheric concentrations of greenhouse gases and, more importantly, related temperature levels that might be considered dangerous.