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

Evaluating a new LISFLOOD-FP formulation with data from the summer 2007 floods in Tewkesbury, UK

Two-dimensional flood inundation modelling is a widely used tool to aid flood risk management. In urban areas, where asset value and population density are greatest, the model spatial resolution required to represent flows through a typical street network (i.e. < 10m) often results in impractical computational cost at the whole city scale. Explicit diffusive storage cell models become very inefficient at such high resolutions, relative to shallow water models, because the stable time step in such schemes scales as a quadratic of resolution. This paper presents the calibration and evaluation of a recently developed new formulation of the LISFLOOD-FP model, where stability is controlled by the Courant–Freidrichs–Levy condition for the shallow water equations, such that, the stable time step instead scales linearly with resolution. The case study used is based on observations during the summer 2007 floods in Tewkesbury, UK. Aerial photography is available for model evaluation on three separate days from the 24th to the 31st of July. The model covered a 3.6 km by 2 km domain and was calibrated using gauge data from high flows during the previous month. The new formulation was benchmarked against the original version of the model at 20 m and 40 m resolutions, demonstrating equally accurate performance given the available validation data but at 67x faster computation time. The July event was then simulated at the 2 m resolution of the available airborne LiDAR DEM. This resulted in a significantly more accurate simulation of the drying dynamics compared to that simulated by the coarse resolution models, although estimates of peak inundation depth were similar.

Formulation of a live bacterial vaccine for stable room temperature storage results in loss of acid, bile and bile salt resistance

Live bacterial vaccines have great promise both as vaccines against enteric pathogens and as heterologous antigen vectors against diverse diseases. Ideally, room temperature stable dry formulations of live bacterial vaccines will allow oral vaccination without cold-chain storage or injections. Attenuated Salmonella can cross the intestinal wall and deliver replicating antigen plus innate immune activation signals directly to the intestinal immune tissues, however the ingested bacteria must survive firstly gastric acid and secondly the antimicrobial defences of the small intestine. We found that the way in which cells are grown prior to formulation markedly affects sensitivity to acid and bile. Using a previously published stable storage formulation that maintained over 10% viability after 56 days storage at room temperature, we found dried samples of an attenuated S. typhimurium vaccine lost acid and bile resistance compared to the same bacteria taken from fresh culture. The stable formulation utilised osmotic preconditioning in defined medium plus elevated salt concentration to induce intracellular trehalose accumulation before drying. Dried bacteria grown in rich media without osmotic preconditioning showed more resistance to bile, but less stability during storage, suggesting a trade-off between bile resistance and stability. Further optimization is needed to produce the ultimate room-temperature stable oral live bacterial vaccine formulation.

Mathematical modelling of decision making processes in construction projects

Many different individuals, who have their own expertise and criteria for decision making, are involved in making decisions on construction projects. Decision-making processes are thus significantly affected by communication, in which a dynamic performance of human intentions leads to unpredictable outcomes. In order to theorise the decision making processes including communication, it is argued here that the decision making processes resemble evolutionary dynamics in terms of both selection and mutation, which can be expressed by the replicator-mutator equation. To support this argument, a mathematical model of decision making has been made from an analogy with evolutionary dynamics, in which there are three variables: initial support rate, business hierarchy, and power of persuasion. On the other hand, a survey of patterns in decision making in construction projects has also been performed through self-administered mail questionnaire to construction practitioners. Consequently, comparison between the numerical analysis of mathematical model and the statistical analysis of empirical data has shown a significant potential of the replicator-mutator equation as a tool to study dynamic properties of intentions in communication.

Formulation and delivery of the bacterial antagonist Bacillus subtilis for management of lentil vascular wilt caused by Fusarium oxysporum f. sp lentis

Different formulations of Bacillus subtilis were prepared using standard laboratory protocols. Bacillus subtilis survived in glucose and talc powders at 8.6 and 7.8 log(10) CFU/g, respectively, for 1 year of storage at room temperature compared with 3.5 log(10) CFU/g on a peat formulation. Glasshouse experiments using soil and seed treatments were conducted to test the efficacy of B. subtilis for protecting lentil against the wilt disease caused by Fusariumoxysporum f. sp. lentis. Seed treatments with formulations of B. subtilis on glucose, talc and peat significantly enhanced its biocontrol activity against Fusarium compared with a treatment in which spores were applied directly to seed. The formulations decreased disease severity by reducing colonization of plants by the pathogen, promoting their growth and increased the dry weight of lentil plants. Of these treatments the glucose and talc-based powder formulations were more effective than the peat formulation and the spore application without a carrier. It was shown that the B. subtilis spores applied with glucose were viable for longer than those applied with other carriers. Seed treatment with these formulated spores is an effective delivery system that can provide a conducive environment for B. subtilis to suppress vascular wilt disease on lentil and has the potential for utilization in commercial field application.

An integrated mathematical model to evaluate nutrient partition in dairy cattle between the animal and its environment

In the past decade, a number of mechanistic, dynamic simulation models of several components of the dairy production system have become available. However their use has been limited due to the detailed technical knowledge and special software required to run them, and the lack of compatibility between models in predicting various metabolic processes in the animal. The first objective of the current study was to integrate the dynamic models of [Brit. J. Nutr. 72 (1994) 679] on rumen function, [J. Anim. Sci. 79 (2001) 1584] on methane production, [J. Anim. Sci. 80 (2002) 2481 on N partition, and a new model of P partition. The second objective was to construct a decision support system to analyse nutrient partition between animal and environment. The integrated model combines key environmental pollutants such as N, P and methane within a nutrient-based feed evaluation system. The model was run under different scenarios and the sensitivity of various parameters analysed. A comparison of predictions from the integrated model with the original simulation models showed an improvement in N excretion since the integrated model uses the dynamic model of [Brit. J. Nutr. 72 (1994) 6791 to predict microbial N, which was not represented in detail in the original model. The integrated model can be used to investigate the degree to which production and environmental objectives are antagonistic, and it may help to explain and understand the complex mechanisms involved at the ruminal and metabolic levels. A part of the integrated model outputs were the forms of N and P in excreta and methane, which can be used as indices of environmental pollution. (C) 2004 Elsevier B.V All rights reserved.

Use and abuse of mathematical models: an illustration from the 2001 foot and mouth disease epidemic in the United Kingdom

Foot and mouth disease (FMD) is a major threat, not only to countries whose economies rely on agricultural exports, but also to industrialised countries that maintain a healthy domestic livestock industry by eliminating major infectious diseases from their livestock populations. Traditional methods of controlling diseases such as FMD require the rapid detection and slaughter of infected animals, and any susceptible animals with which they may have been in contact, either directly or indirectly. During the 2001 epidemic of FMD in the United Kingdom (UK), this approach was supplemented by a culling policy driven by unvalidated predictive models. The epidemic and its control resulted in the death of approximately ten million animals, public disgust with the magnitude of the slaughter, and political resolve to adopt alternative options, notably including vaccination, to control any future epidemics. The UK experience provides a salutary warning of how models can be abused in the interests of scientific opportunism.

Formulating generalised 'goal games' against nature: An illustration from decision-making under uncertainty in agriculture

The games-against-nature approach to the analysis of uncertainty in decision-making relies on the assumption that the behaviour of a decision-maker can be explained by concepts such as maximin, minimax regret, or a similarly defined criterion. In reality, however, these criteria represent a spectrum and, the actual behaviour of a decision-maker is most likely to embody a mixture of such idealisations. This paper proposes that in game-theoretic approach to decision-making under uncertainty, a more realistic representation of a decision-maker's behaviour can be achieved by synthesising games-against-nature with goal programming into a single framework. The proposed formulation is illustrated by using a well-known example from the literature on mathematical programming models for agricultural-decision-making. (c) 2005 Elsevier Inc. All rights reserved.

A mathematical model of the in vitro keratinocyte response to chromium and nickel exposure

A mathematical model describing the main mechanistic processes involved in keratinocyte response to chromium and nickel has been developed and compared to experimental in vitro data. Accounting for the interactions between the metal ions and the keratinocytes, the law of mass action was used to generate ordinary differential equations which predict the time evolution and ion concentration dependency of keratinocyte viability, the amount of metal associated with the keratinocytes and the release of cytokines by the keratinocytes. Good agreement between model predictions and existing experimental data of these endpoints was observed, supporting the use of this model to explore physiochemical parameters that influence the toxicological response of keratinocytes to these two metals.

Understanding post-operative temperature drop in cardiac surgery: a mathematical model

A mathematical model is presented to understand heat transfer processes during the cooling and re-warming of patients during cardiac surgery. Our compartmental model is able to account for many of the qualitative features observed in the cooling of various regions of the body including the central core containing the majority of organs, the rectal region containing the intestines and the outer peripheral region of skin and muscle. In particular, we focus on the issue of afterdrop: a drop in core temperature following patient re-warming, which can lead to serious post-operative complications. Model results for a typical cooling and re-warming procedure during surgery are in qualitative agreement with experimental data in producing the afterdrop effect and the observed dynamical variation in temperature between the core, rectal and peripheral regions. The influence of heat transfer processes and the volume of each compartmental region on the afterdrop effect is discussed. We find that excess fat on the peripheral and rectal regions leads to an increase in the afterdrop effect. Our model predicts that, by allowing constant re-warming after the core temperature has been raised, the afterdrop effect will be reduced.

Overview of mathematical approaches used to model bacterial chemotaxis I: the single cell

Mathematical modeling of bacterial chemotaxis systems has been influential and insightful in helping to understand experimental observations. We provide here a comprehensive overview of the range of mathematical approaches used for modeling, within a single bacterium, chemotactic processes caused by changes to external gradients in its environment. Specific areas of the bacterial system which have been studied and modeled are discussed in detail, including the modeling of adaptation in response to attractant gradients, the intracellular phosphorylation cascade, membrane receptor clustering, and spatial modeling of intracellular protein signal transduction. The importance of producing robust models that address adaptation, gain, and sensitivity are also discussed. This review highlights that while mathematical modeling has aided in understanding bacterial chemotaxis on the individual cell scale and guiding experimental design, no single model succeeds in robustly describing all of the basic elements of the cell. We conclude by discussing the importance of this and the future of modeling in this area.

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller-Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated.

Past, present and future mathematical models for buildings (i)

This is the first of two articles presenting a detailed review of the historical evolution of mathematical models applied in the development of building technology, including conventional buildings and intelligent buildings. After presenting the technical differences between conventional and intelligent buildings, this article reviews the existing mathematical models, the abstract levels of these models, and their links to the literature for intelligent buildings. The advantages and limitations of the applied mathematical models are identified and the models are classified in terms of their application range and goal. We then describe how the early mathematical models, mainly physical models applied to conventional buildings, have faced new challenges for the design and management of intelligent buildings and led to the use of models which offer more flexibility to better cope with various uncertainties. In contrast with the early modelling techniques, model approaches adopted in neural networks, expert systems, fuzzy logic and genetic models provide a promising method to accommodate these complications as intelligent buildings now need integrated technologies which involve solving complex, multi-objective and integrated decision problems.