Very large inverse problems in atmosphere and ocean modelling

For the very large nonlinear dynamical systems that arise in a wide range of physical, biological and environmental problems, the data needed to initialize a numerical forecasting model are seldom available. To generate accurate estimates of the expected states of the system, both current and future, the technique of ‘data assimilation’ is used to combine the numerical model predictions with observations of the system measured over time. Assimilation of data is an inverse problem that for very large-scale systems is generally ill-posed. In four-dimensional variational assimilation schemes, the dynamical model equations provide constraints that act to spread information into data sparse regions, enabling the state of the system to be reconstructed accurately. The mechanism for this is not well understood. Singular value decomposition techniques are applied here to the observability matrix of the system in order to analyse the critical features in this process. Simplified models are used to demonstrate how information is propagated from observed regions into unobserved areas. The impact of the size of the observational noise and the temporal position of the observations is examined. The best signal-to-noise ratio needed to extract the most information from the observations is estimated using Tikhonov regularization theory. Copyright © 2005 John Wiley & Sons, Ltd.

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

A comparison of two methods for developing the linearization of a shallow-water model

We develop the linearization of a semi-implicit semi-Lagrangian model of the one-dimensional shallow-water equations using two different methods. The usual tangent linear model, formed by linearizing the discrete nonlinear model, is compared with a model formed by first linearizing the continuous nonlinear equations and then discretizing. Both models are shown to perform equally well for finite perturbations. However, the asymptotic behaviour of the two models differs as the perturbation size is reduced. This leads to difficulties in showing that the models are correctly coded using the standard tests. To overcome this difficulty we propose a new method for testing linear models, which we demonstrate both theoretically and numerically. © Crown copyright, 2003. Royal Meteorological Society

Progress on “Changing coastlines: data assimilation for morphodynamic prediction and predictability”

The task of assessing the likelihood and extent of coastal flooding is hampered by the lack of detailed information on near-shore bathymetry. This is required as an input for coastal inundation models, and in some cases the variability in the bathymetry can impact the prediction of those areas likely to be affected by flooding in a storm. The constant monitoring and data collection that would be required to characterise the near-shore bathymetry over large coastal areas is impractical, leaving the option of running morphodynamic models to predict the likely bathymetry at any given time. However, if the models are inaccurate the errors may be significant if incorrect bathymetry is used to predict possible flood risks. This project is assessing the use of data assimilation techniques to improve the predictions from a simple model, by rigorously incorporating observations of the bathymetry into the model, to bring the model closer to the actual situation. Currently we are concentrating on Morecambe Bay as a primary study site, as it has a highly dynamic inter-tidal zone, with changes in the course of channels in this zone impacting the likely locations of flooding from storms. We are working with SAR images, LiDAR, and swath bathymetry to give us the observations over a 2.5 year period running from May 2003 – November 2005. We have a LiDAR image of the entire inter-tidal zone for November 2005 to use as validation data. We have implemented a 3D-Var data assimilation scheme, to investigate the improvements in performance of the data assimilation compared to the previous scheme which was based on the optimal interpolation method. We are currently evaluating these different data assimilation techniques, using 22 SAR data observations. We will also include the LiDAR data and swath bathymetry to improve the observational coverage, and investigate the impact of different types of observation on the predictive ability of the model. We are also assessing the ability of the data assimilation scheme to recover the correct bathymetry after storm events, which can dramatically change the bathymetry in a short period of time.

Endgame table testing of studies - II

The van der Heijden Studies Database (Version III) has been reviewed to identify 'Win Studies' with sub-7-man positions in the main line which are not wins for White. Some studies were faulted, A number for the first time: 21 of the more interesting escapes by Black are highlighted, themed and discussed.

Developmental changes in the germinability, desiccation tolerance, hardseededness, and longevity of individual seeds of Trifolium ambiguum

Background and Aims: Using two parental clones of outcrossing Trifolium ambiguum as a potential model system, we examined how during seed development the maternal parent, number of seeds per pod, seed position within the pod, and pod position within the inflorescence influenced individual seed fresh weight, dry weight, water content, germinability, desiccation tolerance, hardseededness, and subsequent longevity of individual seeds. Methods: Near simultaneous, manual reciprocal crosses were carried out between clonal lines for two experiments. Infructescences were harvested at intervals during seed development. Each individual seed was weighed and then used to determine dry weight or one of the physiological behaviour traits. Key Results: Whilst population mass maturity was reached at 33–36 days after pollination (DAP), seed-to-seed variation in maximum seed dry weight, when it was achieved, and when maturation drying commenced, was considerable. Individual seeds acquired germinability between 14 and 44 DAP, desiccation tolerance between 30 and 40 DAP, and the capability to become hardseeded between 30 and 47 DAP. The time for viability to fall to 50 % (p50) at 60 % relative humidity and 45 °C increased between 36 and 56 DAP, when the seed coats of most individuals had become dark orange, but declined thereafter. Individual seed f. wt at harvest did not correlate with air-dry storage survival period. Analysing survival data for cohorts of seeds reduced the standard deviation of the normal distribution of seed deaths in time, but no sub-population showed complete uniformity of survival period. Conclusions: Variation in individual seed behaviours within a developing population is inherent and inevitable. In this outbreeder, there is significant variation in seed longevity which appears dependent on embryo genotype with little effect of maternal genotype or architectural factors.