A simple column model to explore anticipated problems in variational assimilation of satellite observations

We investigate a simplified form of variational data assimilation in a fully nonlinear framework with the aim of extracting dynamical development information from a sequence of observations over time. Information on the vertical wind profile, w(z ), and profiles of temperature, T (z , t), and total water content, qt (z , t), as functions of height, z , and time, t, are converted to brightness temperatures at a single horizontal location by defining a two-dimensional (vertical and time) variational assimilation testbed. The profiles of T and qt are updated using a vertical advection scheme. A basic cloud scheme is used to obtain the fractional cloud amount and, when combined with the temperature field, this information is converted into a brightness temperature, using a simple radiative transfer scheme. It is shown that our model exhibits realistic behaviour with regard to the prediction of cloud, but the effects of nonlinearity become non-negligible in the variational data assimilation algorithm. A careful analysis of the application of the data assimilation scheme to this nonlinear problem is presented, the salient difficulties are highlighted, and suggestions for further developments are discussed.

The role of location in knowledge creation and diffusion: evidence of centripetal and centrifugal forces in the City of London financial services agglomeration

In this paper we report on a major empirical study of centripetal and centrifugal forces in the City of London financial services agglomeration. The study sheds light on (1) the manner and magnitude of firm interaction in the agglomeration; (2) the characteristics of the agglomeration that aid the competitiveness of incumbent firms; and (3) the problems associated with agglomeration. In addressing these issues, we use the data to (1) test emerging theory that explains the high productivity and innovation of agglomerations in terms of their ability to generate and diffuse knowledge; and (2) evaluate the ‘end of geography’ thesis.

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.

An integral equation method for a boundary value problem arising in unsteady water wave problems

In this paper we consider the 2D Dirichlet boundary value problem for Laplace’s equation in a non-locally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragmen-Lindelof arguments. The main result is to propose a boundary integral equation formulation, to prove equivalence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalisation of the Fredholm alternative in Arens et al [J. Int. Equ. Appl. 15 (2003) pp. 1-35]. This then leads to an existence proof for the boundary value problem. Keywords. Boundary integral equation method, Water waves, Laplace’s

Consistent Dirichlet Boundary Conditions for Numerical Solution of Moving Boundary Problems

We consider the imposition of Dirichlet boundary conditions in the finite element modelling of moving boundary problems in one and two dimensions for which the total mass is prescribed. A modification of the standard linear finite element test space allows the boundary conditions to be imposed strongly whilst simultaneously conserving a discrete mass. The validity of the technique is assessed for a specific moving mesh finite element method, although the approach is more general. Numerical comparisons are carried out for mass-conserving solutions of the porous medium equation with Dirichlet boundary conditions and for a moving boundary problem with a source term and time-varying mass.

Boundary value problems for third-order linear PDEs in time-dependent domains

We present the extension of a methodology to solve moving boundary value problems from the second-order case to the case of the third-order linear evolution PDE qt + qxxx = 0. This extension is the crucial step needed to generalize this methodology to PDEs of arbitrary order. The methodology is based on the derivation of inversion formulae for a class of integral transforms that generalize the Fourier transform and on the analysis of the global relation associated with the PDE. The study of this relation and its inversion using the appropriate generalized transform are the main elements of the proof of our results.

An interpretation of baroclinic initial value problems: results for simple basic states with nonzero interior PV gradients

In the Eady model, where the meridional potential vorticity (PV) gradient is zero, perturbation energy growth can be partitioned cleanly into three mechanisms: (i) shear instability, (ii) resonance, and (iii) the Orr mechanism. Shear instability involves two-way interaction between Rossby edge waves on the ground and lid, resonance occurs as interior PV anomalies excite the edge waves, and the Orr mechanism involves only interior PV anomalies. These mechanisms have distinct implications for the structural and temporal linear evolution of perturbations. Here, a new framework is developed in which the same mechanisms can be distinguished for growth on basic states with nonzero interior PV gradients. It is further shown that the evolution from quite general initial conditions can be accurately described (peak error in perturbation total energy typically less than 10%) by a reduced system that involves only three Rossby wave components. Two of these are counterpropagating Rossby waves—that is, generalizations of the Rossby edge waves when the interior PV gradient is nonzero—whereas the other component depends on the structure of the initial condition and its PV is advected passively with the shear flow. In the cases considered, the three-component model outperforms approximate solutions based on truncating a modal or singular vector basis.

Cue combination for 3D location judgements

Cue combination rules have often been applied to the perception of surface shape but not to judgements of object location. Here, we used immersive virtual reality to explore the relationship between different cues to distance. Participants viewed a virtual scene and judged the change in distance of an object presented in two intervals, where the scene changed in size between intervals (by a factor of between 0.25 and 4). We measured thresholds for detecting a change in object distance when there were only 'physical' (stereo and motion parallax) or 'texture-based' cues (independent of the scale of the scene) and used these to predict biases in a distance matching task. Under a range of conditions, in which the viewing distance and position of the tarte relative to other objects was varied, the ration of 'physical' to 'texture-based' thresholds was a good predictor of biases in the distance matching task. The cue combination approach, which successfully accounts for our data, relies on quite different principles from those underlying geometric reconstruction.

Evolving graphs: dynamical models, inverse problems and propagation

Applications such as neuroscience, telecommunication, online social networking, transport and retail trading give rise to connectivity patterns that change over time. In this work, we address the resulting need for network models and computational algorithms that deal with dynamic links. We introduce a new class of evolving range-dependent random graphs that gives a tractable framework for modelling and simulation. We develop a spectral algorithm for calibrating a set of edge ranges from a sequence of network snapshots and give a proof of principle illustration on some neuroscience data. We also show how the model can be used computationally and analytically to investigate the scenario where an evolutionary process, such as an epidemic, takes place on an evolving network. This allows us to study the cumulative effect of two distinct types of dynamics.

Aqueous geochemistry and oxygen isotope compositions of acid mine drainage from the Río Tinto, SW Spain, highlight inconsistencies in current models

The Rio Tinto river in SW Spain is a classic example of acid mine drainage and the focus of an increasing amount of research including environmental geochemistry, extremophile microbiology and Mars-analogue studies. Its 5000-year mining legacy has resulted in a wide range of point inputs including spoil heaps and tunnels draining underground workings. The variety of inputs and importance of the river as a research site make it an ideal location for investigating sulphide oxidation mechanisms at the field scale. Mass balance calculations showed that pyrite oxidation accounts for over 93% of the dissolved sulphate derived from sulphide oxidation in the Rio Tinto point inputs. Oxygen isotopes in water and sulphate were analysed from a variety of drainage sources and displayed delta O-18((SO4-H2O)) values from 3.9 to 13.6 parts per thousand, indicating that different oxidation pathways occurred at different sites within the catchment. The most commonly used approach to interpreting field oxygen isotope data applies water and oxygen fractionation factors derived from laboratory experiments. We demonstrate that this approach cannot explain high delta O-18((SO4-H2O)) values in a manner that is consistent with recent models of pyrite and sulphoxyanion oxidation. In the Rio Tinto, high delta O-18((SO4-H2O)) values (11.2-13.6 parts per thousand) occur in concentrated (Fe = 172-829 mM), low pH (0.88-1.4), ferrous iron (68-91% of total Fe) waters and are most simply explained by a mechanism involving a dissolved sulphite intermediate, sulphite-water oxygen equilibrium exchange and finally sulphite oxidation to sulphate with O-2. In contrast, drainage from large waste blocks of acid volcanic tuff with pyritiferous veins also had low pH (1.7). but had a low delta O-18((SO4-H2O)) value of 4.0 parts per thousand and high concentrations of ferric iron (Fe(III) = 185 mM, total Fe = 186 mM), suggesting a pathway where ferric iron is the primary oxidant, water is the primary source of oxygen in the sulphate and where sulphate is released directly from the pyrite surface. However, problems remain with the sulphite-water oxygen exchange model and recommendations are therefore made for future experiments to refine our understanding of oxygen isotopes in pyrite oxidation. (C) 2009 Elsevier B.V. All rights reserved.

Problems with determining the particle size distribution of chalk soil and some of their implications

Particle size distribution (psd) is one of the most important features of the soil because it affects many of its other properties, and it determines how soil should be managed. To understand the properties of chalk soil, psd analyses should be based on the original material (including carbonates), and not just the acid-resistant fraction. Laser-based methods rather than traditional sedimentation methods are being used increasingly to determine particle size to reduce the cost of analysis. We give an overview of both approaches and the problems associated with them for analyzing the psd of chalk soil. In particular, we show that it is not appropriate to use the widely adopted 8 pm boundary between the clay and silt size fractions for samples determined by laser to estimate proportions of these size fractions that are equivalent to those based on sedimentation. We present data from field and national-scale surveys of soil derived from chalk in England. Results from both types of survey showed that laser methods tend to over-estimate the clay-size fraction compared to sedimentation for the 8 mu m clay/silt boundary, and we suggest reasons for this. For soil derived from chalk, either the sedimentation methods need to be modified or it would be more appropriate to use a 4 pm threshold as an interim solution for laser methods. Correlations between the proportions of sand- and clay-sized fractions, and other properties such as organic matter and volumetric water content, were the opposite of what one would expect for soil dominated by silicate minerals. For water content, this appeared to be due to the predominance of porous, chalk fragments in the sand-sized fraction rather than quartz grains, and the abundance of fine (<2 mu m) calcite crystals rather than phyllosilicates in the clay-sized fraction. This was confirmed by scanning electron microscope (SEM) analyses. "Of all the rocks with which 1 am acquainted, there is none whose formation seems to tax the ingenuity of theorists so severely, as the chalk, in whatever respect we may think fit to consider it". Thomas Allan, FRS Edinburgh 1823, Transactions of the Royal Society of Edinburgh. (C) 2009 Natural Environment Research Council (NERC) Published by Elsevier B.V. All rights reserved.

The Dubawnt Lake palaeo-ice stream: evidence for dynamic ics sheet behaviour on the Canadian Shield and insights regarding the controls on ice-stream location and vigour

We report evidence for a major ice stream that operated over the northwestern Canadian Shield in the Keewatin Sector of the Laurentide Ice Sheet during the last deglaciation 9000-8200 (uncalibrated) yr BP. It is reconstructed at 450 km in length, 140 km in width, and had an estimated catchment area of 190000 km. Mapping from satellite imagery reveals a suite of bedforms ('flow-set') characterized by a highly convergent onset zone, abrupt lateral margins, and where flow was presumed to have been fastest, a remarkably coherent pattern of mega-scale glacial lineations with lengths approaching 13 km and elongation ratios in excess of 40:1. Spatial variations in bedform elongation within the flow-set match the expected velocity field of a terrestrial ice stream. The flow pattern does not appear to be steered by topography and its location on the hard bedrock of the Canadian Shield is surprising. A soft sedimentary basin may have influenced ice-stream activity by lubricating the bed over the downstream crystalline bedrock, but it is unlikely that it operated over a pervasively deforming till layer. The location of the ice stream challenges the view that they only arise in deep bedrock troughs or over thick deposits of 'soft' fine-grained sediments. We speculate that fast ice flow may have been triggered when a steep ice sheet surface gradient with high driving stresses contacted a proglacial lake. An increase in velocity through calving could have propagated fast ice flow upstream (in the vicinity of the Keewatin Ice Divide) through a series of thermomechanical feedback mechanisms. It exerted a considerable impact on the Laurentide Ice Sheet, forcing the demise of one of the last major ice centres.