Conditioning of the weak-constraint variational data assimilation problem for numerical weather prediction

4-Dimensional Variational Data Assimilation (4DVAR) assimilates observations through the minimisation of a least-squares objective function, which is constrained by the model flow. We refer to 4DVAR as strong-constraint 4DVAR (sc4DVAR) in this thesis as it assumes the model is perfect. Relaxing this assumption gives rise to weak-constraint 4DVAR (wc4DVAR), leading to a different minimisation problem with more degrees of freedom. We consider two wc4DVAR formulations in this thesis, the model error formulation and state estimation formulation. The 4DVAR objective function is traditionally solved using gradient-based iterative methods. The principle method used in Numerical Weather Prediction today is the Gauss-Newton approach. This method introduces a linearised `inner-loop' objective function, which upon convergence, updates the solution of the non-linear `outer-loop' objective function. This requires many evaluations of the objective function and its gradient, which emphasises the importance of the Hessian. The eigenvalues and eigenvectors of the Hessian provide insight into the degree of convexity of the objective function, while also indicating the difficulty one may encounter while iterative solving 4DVAR. The condition number of the Hessian is an appropriate measure for the sensitivity of the problem to input data. The condition number can also indicate the rate of convergence and solution accuracy of the minimisation algorithm. This thesis investigates the sensitivity of the solution process minimising both wc4DVAR objective functions to the internal assimilation parameters composing the problem. We gain insight into these sensitivities by bounding the condition number of the Hessians of both objective functions. We also precondition the model error objective function and show improved convergence. We show that both formulations' sensitivities are related to error variance balance, assimilation window length and correlation length-scales using the bounds. We further demonstrate this through numerical experiments on the condition number and data assimilation experiments using linear and non-linear chaotic toy models.

Conditioning and preconditioning of the optimal state estimation problem

Optimal state estimation is a method that requires minimising a weighted, nonlinear, least-squares objective function in order to obtain the best estimate of the current state of a dynamical system. Often the minimisation is non-trivial due to the large scale of the problem, the relative sparsity of the observations and the nonlinearity of the objective function. To simplify the problem the solution is often found via a sequence of linearised objective functions. The condition number of the Hessian of the linearised problem is an important indicator of the convergence rate of the minimisation and the expected accuracy of the solution. In the standard formulation the convergence is slow, indicating an ill-conditioned objective function. A transformation to different variables is often used to ameliorate the conditioning of the Hessian by changing, or preconditioning, the Hessian. There is only sparse information in the literature for describing the causes of ill-conditioning of the optimal state estimation problem and explaining the effect of preconditioning on the condition number. This paper derives descriptive theoretical bounds on the condition number of both the unpreconditioned and preconditioned system in order to better understand the conditioning of the problem. We use these bounds to explain why the standard objective function is often ill-conditioned and why a standard preconditioning reduces the condition number. We also use the bounds on the preconditioned Hessian to understand the main factors that affect the conditioning of the system. We illustrate the results with simple numerical experiments.

The effects of the insecticide chlorpyrifos on spider and Collembola communities

The effects of chlorpyrifos on aquatic systems are well documented. However, the consequences of the pesticide on soil food webs are poorly understood. In this field study, we hypothesised that the addition of a soil insecticide to an area of upland grassland would impact spider and Collembola communities by decreasing numbers of spiders, consequently, causing an increase in detritivore numbers and diversity. Chlorpyrifos was added to plots on an upland grassland in a randomised block design. Populations of Collembola and spiders were sampled by means of pitfall traps (activity density) and identified to species. Twelve species of Collembola were identified from the insecticide-treated and control plots. Species diversity, richness and evenness were all reduced in the chlorpyrifos plots, although the total number of Collembola increased ten-fold despite the abundance of some spider species being reduced. The dominant collembolan in the insecticide-treated plots was Ceratophysella denticulata, accounting for over 95% of the population. Forty-three species of spider were identified. There were a reduced number of spiders in insecticide-treated plots due mainly to a lower number of the linyphiid, Tiso vagans. However, there was no significant difference in spider diversity between the control and insecticide treatments. We discuss possible explanations for the increase in abundance of one collembolan species in response to chlorpyrifos and the consequences of this. The study emphasises the importance of understanding the effects of soil management practices on soil biodiversity, which is under increasing pressure from land development and food production. It also highlights the need for identification of soil invertebrates to an 'appropriate' taxonomic level for biodiversity estimates. (C) 2007 Elsevier GrnbH. All rights reserved.

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.

The effect of the year of wheat variety release on productivity and stability of performance on two organic and two non-organic farms

Nineteen wheat cultivars, released from 1934 to 2000, were grown at two organic and two non-organic sites in each of 3 years. Assessments included grain yield, grain protein concentration, protein yield, disease incidence and green leaf area. The superiority of each cultivar (the sum of the squares of the differences between its mean in each environment and the mean of the best cultivar there, divided by twice the number of environments; CS) was calculated for yield, grain protein concentration and protein yield, and ranked in each environment. The yield and grain protein concentration CS were more closely correlated with cultivar release date at the non-organic sites than at organic sites. This difference may be attributed to higher yield levels with larger differences among cultivars at the non-organic sites, rather than to improved stability (i.e. similar ranks) across sites. The significant difference in the correlation of protein yield CS and cultivar age between organic and non-organic sites would support evidence that the ability to take up mineral nitrogen (N) compared to soil N has been a component of the selection conditions of more modern cultivars (released after 1989). This is supported by assessment of green leaf area (GLA), where more modern cultivars in the non-organic systems had greater late-season GLA, a trend that was not identified in organic conditions. This effect could explain the poor correlation between age and protein yield CS in organic compared to non-organic conditions where modern cultivars are selected to benefit from later nitrogen (N) availability which includes the spring nitrogen applications tailored to coincide with peak crop demand. Under organic management, N release is largely based on the breakdown of fertility-building crops incorporated (ploughed-in) in the previous autumn. The release of nutrients from these residues is dependent on the soil conditions, which includes temperature and microbial populations, in addition to the potential leaching effect of high winter rainfall in the UK. In organic cereal crops, early resource capture is a major advantage for maximizing the utilization of nutrients from residue breakdown. It is concluded that selection of cultivars under conditions of high agrochemical inputs selects for cultivars that yield well under maximal conditions in terms of nutrient availability and pest, disease and weed control. The selection conditions for breeding have a tendency to select cultivars which perform relatively better in non-organic compared to organic systems.

Detecting the number of signals using antenna array: a single threshold solution

A new approach is presented to identify the number of incoming signals in antenna array processing. The new method exploits the inherent properties existing in the noise eigenvalues of the covariance matrix of the array output. A single threshold has been established concerning information about the signal and noise strength, data length, and array size. When the subspace-based algorithms are adopted the computation cost of the signal number detector can almost be neglected. The performance of the threshold is robust against low SNR and short data length.

A molecular, morphological and cytological investigation of the identity of non-native Ludwigia (Onagraceae) populations in Britain

A combined molecular, morphological and cytological analysis was used to study the identity and number of species of Ludwigia section Oligospermum occurring in British waterways. Only one taxon was identified for which the name L. grandiflora subsp. hexapetala (Hook. & Arn.) G.L. Nesom & Kartesz is preferred. A chromosome count of 2n = 80 was made for all plants tested and DNA evidence demonstrates that at least two clones are present in Britain. Morphological characters to differentiate L.grandiflora subsp. hexapetala and L. peploides subsp. montevidensis (Spreng.) P.H. Raven are provided. Though the production of fruit in Britain by apparently isolated populations is noted, repeated introduction into the wild from gardens is judged to be primarily responsible for the British distribution of the taxon. Legislative implications are considered and an amendment to Schedule 9 of the UK Wildlife and Countryside Act (1981) is strongly advocated.

Social equality in the number of choice options is represented in the ventromedial prefrontal cortex

A distinct aspect of the sense of fairness in humans is that we care not only about equality in material rewards but also about equality in non-material values. One such value is the opportunity to choose freely among many options, often regarded as a fundamental right to economic freedom. In modern developed societies, equal opportunities in work, living, and lifestyle are enforced by anti-discrimination laws. Despite the widespread endorsement of equal opportunity, no studies have explored how people assign value to it. We used functional magnetic resonance imaging to identify the neural substrates for subjective valuation of equality in choice opportunity. Participants performed a two-person choice task in which the number of choices available was varied across trials independently of choice outcomes. By using this procedure, we manipulated the degree of equality in choice opportunity between players and dissociated it from the value of reward outcomes and their equality. We found that activation in the ventromedial prefrontal cortex tracked the degree to which the number of options between the two players was equal. In contrast, activation in the ventral striatum tracked the number of options available to participants themselves but not the equality between players. Our results demonstrate that the vmPFC, a key brain region previously implicated in the processing of social values, is also involved in valuation of equality in choice opportunity between individuals. These findings may provide valuable insight into the human ability to value equal opportunity, a characteristic long emphasized in politics, economics, and philosophy.

Creating a non-word list to match 226 of the Snodgrass standardised picture set

Creating non-word lists is a necessary but time consuming exercise often needed when conducting behavioural language tasks involving lexical decision-making or non-word reading. The following article describes the process whereby we created a list of 226 non-words matching 226 of the Snodgrass picture set (Snodgrass & Vanderwart, 1980).The non-words were matched for number of syllables, stress pattern, number of phonemes, bigram count and presence and location of the target sound when relevant.

High-precision measurements of the co-polar correlation coefficient: non-Gaussian errors and retrieval of the dispersion parameter µ in rainfall

The co-polar correlation coefficient (ρhv) has many applications, including hydrometeor classification, ground clutter and melting layer identification, interpretation of ice microphysics and the retrieval of rain drop size distributions (DSDs). However, we currently lack the quantitative error estimates that are necessary if these applications are to be fully exploited. Previous error estimates of ρhv rely on knowledge of the unknown "true" ρhv and implicitly assume a Gaussian probability distribution function of ρhv samples. We show that frequency distributions of ρhv estimates are in fact highly negatively skewed. A new variable: L = -log10(1 - ρhv) is defined, which does have Gaussian error statistics, and a standard deviation depending only on the number of independent radar pulses. This is verified using observations of spherical drizzle drops, allowing, for the first time, the construction of rigorous confidence intervals in estimates of ρhv. In addition, we demonstrate how the imperfect co-location of the horizontal and vertical polarisation sample volumes may be accounted for. The possibility of using L to estimate the dispersion parameter (µ) in the gamma drop size distribution is investigated. We find that including drop oscillations is essential for this application, otherwise there could be biases in retrieved µ of up to ~8. Preliminary results in rainfall are presented. In a convective rain case study, our estimates show µ to be substantially larger than 0 (an exponential DSD). In this particular rain event, rain rate would be overestimated by up to 50% if a simple exponential DSD is assumed.

Excluding exceptions: Housing non-affordability and the oppression of environmental sustainability?

This paper explores the provision of homes for less wealthy households in rural England. By allowing 'exceptions' to UK planning law to provide low-income housing for local residents, the national government seeks to secure dwellings for the less wealthy and so sustain socially mixed rural villages. This paper explores how the production of homes through the exception policy is not conducive to the construction of many new houses. The particular emphasis in the paper is on how responsible agents are discouraged from being more active in erecting new village homes for low-income households. Empirically, the paper draws on documents, interviews and a social survey in the counties of Bedfordshire, Cambridgeshire and Norfolk to investigate the process of delivering rural exception homes. It is concluded that, despite Government assertions that a socially mixed countryside is desirable, the decision-making criteria that dominate the worldviews of agents in social housing provision work against this outcome. (c) 2005 Elsevier Ltd. All rights reserved.

Driving places - Marc Augé, non-places, and the geographies of England's M1 motorway

In this article I provide a critical account of the 'placing' of England's M1 motor-way. I start by critiquing Marc Auge's anthropological writings on 'non-places' which have provided a common point of reference for academics discussing spaces of travel, consumption and exchange in the contemporary world. I argue that Auge's ethnology of supermodernity results in a rather partial account of these sites, that he overstates the novelty of contemporary experiences of these spaces, and that he fails to acknowledge the heterogeneity and materiality of the social networks bound up with the production of non-places/places. I suggest that, rather than focusing on the presences and absences associated with the polarities of place and non-place, academics should examine the multiple, partial, dynamic and relational 'placings' which arise through the diverse performances and movements associated with travel, consumption and exchange. I then trace the topologies of England's M1 motorway, examining some of the different ways in which the motorway has been assembled, performed and placed over the past 45 years.