Efficient non-linear data assimilation in geophysical fluid dynamics

New ways of combining observations with numerical models are discussed in which the size of the state space can be very large, and the model can be highly nonlinear. Also the observations of the system can be related to the model variables in highly nonlinear ways, making this data-assimilation (or inverse) problem highly nonlinear. First we discuss the connection between data assimilation and inverse problems, including regularization. We explore the choice of proposal density in a Particle Filter and show how the ’curse of dimensionality’ might be beaten. In the standard Particle Filter ensembles of model runs are propagated forward in time until observations are encountered, rendering it a pure Monte-Carlo method. In large-dimensional systems this is very inefficient and very large numbers of model runs are needed to solve the data-assimilation problem realistically. In our approach we steer all model runs towards the observations resulting in a much more efficient method. By further ’ensuring almost equal weight’ we avoid performing model runs that are useless in the end. Results are shown for the 40 and 1000 dimensional Lorenz 1995 model.

The Mathematics of Control Theory

This text contains papers presented at the Institute of Mathematics and its Applications Conference on Control Theory, held at the University of Strathclyde in Glasgow. The contributions cover a wide range of topics of current interest to theoreticians and practitioners including algebraic systems theory, nonlinear control systems, adaptive control, robustness issues, infinite dimensional systems, applications studies and connections to mathematical aspects of information theory and data-fusion.

Ensemble clustering in deterministic ensemble Kalman filters

Ensemble clustering (EC) can arise in data assimilation with ensemble square root filters (EnSRFs) using non-linear models: an M-member ensemble splits into a single outlier and a cluster of M−1 members. The stochastic Ensemble Kalman Filter does not present this problem. Modifications to the EnSRFs by a periodic resampling of the ensemble through random rotations have been proposed to address it. We introduce a metric to quantify the presence of EC and present evidence to dispel the notion that EC leads to filter failure. Starting from a univariate model, we show that EC is not a permanent but transient phenomenon; it occurs intermittently in non-linear models. We perform a series of data assimilation experiments using a standard EnSRF and a modified EnSRF by a resampling though random rotations. The modified EnSRF thus alleviates issues associated with EC at the cost of traceability of individual ensemble trajectories and cannot use some of algorithms that enhance performance of standard EnSRF. In the non-linear regimes of low-dimensional models, the analysis root mean square error of the standard EnSRF slowly grows with ensemble size if the size is larger than the dimension of the model state. However, we do not observe this problem in a more complex model that uses an ensemble size much smaller than the dimension of the model state, along with inflation and localisation. Overall, we find that transient EC does not handicap the performance of the standard EnSRF.

Indian summer monsoon variations could have affected the early Holocene woodland expansion in the Near East

Postglacial expansion of deciduous oak woodlands of the Zagros—Anti-Taurus Mountains, a major biome of the Near East, was delayed until the middle Holocene at ~6300 cal. yr BP. The current hypotheses explain this delay as a consequence of a regional aridity during the early Holocene, slow migration rates of forest trees, and/or a long history of land use and agro-pastoralism in this region. In the present paper, support is given to a hypothesis that suggests different precipitation seasonalities during the early Holocene compared with the late Holocene. The oak species of the Zagros—Anti-Taurus Mts, particularly Quercus brantii Lindl., are strongly dependent on spring precipitation for regeneration and are sensitive to a long dry season. Detailed analysis of modern atmospheric circulation patterns in SW Asia during the late spring suggests that the Indian Summer Monsoon (ISM) intensification can modify the amount of late spring and/or early summer rainfall in western/northwestern Iran and eastern Anatolia, which could in turn have controlled the development of the Zagros—Anti-Taurus deciduous oak woodlands. During the early Holocene, the northwestward shift of the Inter-Tropical Convergence Zone (ITCZ) could have displaced the subtropical anticyclonic belt or associated high pressure ridges to the northwest. The latter could, in turn, have prevented the southeastward penetration of low pressure systems originating from the North Atlantic and Black Sea regions. Such atmospheric configuration could have reduced or eliminated the spring precipitation creating a typical Mediterranean continental climate characterized by winter-dominated precipitation. This scenario highlights the complexity of biome response to climate system interactions in transitional climatic and biogeographical regions.

A comparative review of dimension reduction methods in approximate Bayesian computation

Approximate Bayesian computation (ABC) methods make use of comparisons between simulated and observed summary statistics to overcome the problem of computationally intractable likelihood functions. As the practical implementation of ABC requires computations based on vectors of summary statistics, rather than full data sets, a central question is how to derive low-dimensional summary statistics from the observed data with minimal loss of information. In this article we provide a comprehensive review and comparison of the performance of the principal methods of dimension reduction proposed in the ABC literature. The methods are split into three nonmutually exclusive classes consisting of best subset selection methods, projection techniques and regularization. In addition, we introduce two new methods of dimension reduction. The first is a best subset selection method based on Akaike and Bayesian information criteria, and the second uses ridge regression as a regularization procedure. We illustrate the performance of these dimension reduction techniques through the analysis of three challenging models and data sets.

An exploration of the equivalent weights particle filter

Particle filters are fully non-linear data assimilation techniques that aim to represent the probability distribution of the model state given the observations (the posterior) by a number of particles. In high-dimensional geophysical applications the number of particles required by the sequential importance resampling (SIR) particle filter in order to capture the high probability region of the posterior, is too large to make them usable. However particle filters can be formulated using proposal densities, which gives greater freedom in how particles are sampled and allows for a much smaller number of particles. Here a particle filter is presented which uses the proposal density to ensure that all particles end up in the high probability region of the posterior probability density function. This gives rise to the possibility of non-linear data assimilation in large dimensional systems. The particle filter formulation is compared to the optimal proposal density particle filter and the implicit particle filter, both of which also utilise a proposal density. We show that when observations are available every time step, both schemes will be degenerate when the number of independent observations is large, unlike the new scheme. The sensitivity of the new scheme to its parameter values is explored theoretically and demonstrated using the Lorenz (1963) model.

Reduced-order dynamics of the Martian atmospheric dynamics

In this paper we explore the possibility of deriving low-dimensional models of the dynamics of the Martian atmosphere. The analysis consists of a Proper Orthogonal Decomposition (POD) of the atmospheric streamfunction after first decomposing the vertical structure with a set of eigenmodes. The vertical modes were obtained from the quasi-geostrophic vertical structure equation. The empirical orthogonal functions (EOFs) were optimized to represent the atmospheric total energy. The total energy was used as the criterion to retain those modes with large energy content and discard the rest. The principal components (PCs) were analysed by means of Fourier analysis, so that the dominant frequencies could be identified. It was possible to observe the strong influence of the diurnal cycle and to identify the motion and vacillation of baroclinic waves.

Optimal steady motions for oriented vehicles

Motivated by the motion planning problem for oriented vehicles travelling in a 3-Dimensional space; Euclidean space E3, the sphere S3 and Hyperboloid H3. For such problems the orientation of the vehicle is naturally represented by an orthonormal frame over a point in the underlying manifold. The orthonormal frame bundles of the space forms R3,S3 and H3 correspond with their isometry groups and are the Euclidean group of motion SE(3), the rotation group SO(4) and the Lorentzian group SO(1; 3) respectively. Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. For constant twist motions or helical motions, the corresponding curves g(t) 2 SE(3) are given in closed form by using the well known Rodrigues’ formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw/helical motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

Gaussian processes autoencoder for dimensionality reduction

Learning low dimensional manifold from highly nonlinear data of high dimensionality has become increasingly important for discovering intrinsic representation that can be utilized for data visualization and preprocessing. The autoencoder is a powerful dimensionality reduction technique based on minimizing reconstruction error, and it has regained popularity because it has been efficiently used for greedy pretraining of deep neural networks. Compared to Neural Network (NN), the superiority of Gaussian Process (GP) has been shown in model inference, optimization and performance. GP has been successfully applied in nonlinear Dimensionality Reduction (DR) algorithms, such as Gaussian Process Latent Variable Model (GPLVM). In this paper we propose the Gaussian Processes Autoencoder Model (GPAM) for dimensionality reduction by extending the classic NN based autoencoder to GP based autoencoder. More interestingly, the novel model can also be viewed as back constrained GPLVM (BC-GPLVM) where the back constraint smooth function is represented by a GP. Experiments verify the performance of the newly proposed model.

The effects of dairy management and processing on quality characteristics of milk and dairy products

Studies within the QLIF project reviewed in this article suggest that organic or low-input management is more likely to result in milk with fatty acid profiles that are higher in α-linolenic acid and/or beneficial isomers of conjugated linoleic acid and antioxidants with up to a 2.5-fold increase in some cases, relative to milk from conventional production. These advantages are preserved during processing, resulting in elevated contents or concentrations of these constituents in processed dairy products of organic or low input origin. Much of the literature suggests that these benefits are very likely to be a result of a greater reliance on forages in the dairy diets (especially grazed grass). Since the adoption of alternative breeds or crosses is often an integral part sustaining these low-input systems, it is not possible to rule out an interaction with genotype in these monitored herds. The results suggest that milk fat composition with respect to human health can be optimized by exploiting grazing in the diet of dairy cows. However, in many European regions this may not be possible due to extremes in temperature, soil moisture levels or both. In such cases milk quality can be maintained by the inclusion of oil seeds in the dairy diets.

LOFZY 2005 - Field experiment on Cyclones over the Norwegian Sea: meteorological measurements of the research aircraft Falcon, 23 autonomous water buoys and radiosoundings at the research vessel Celtic Explorer

The field campaign LOFZY 2005 (LOFoten ZYklonen, engl.: Cyclones) was carried out in the frame of Collaborative Research Centre 512, which deals with low-pressure systems (cyclones) and the climate system of the North Atlantic. Cyclones are of special interest due to their influence on the interaction between atmosphere and ocean. Cyclone activity in the northern part of the Atlantic Ocean is notably high and is of particular importance for the entire Atlantic Ocean. An area of maximum precipitation exists in front of the Norwegian Lofoten islands. One aim of the LOFZY field campaign was to clarify the role cyclones play in the interaction of ocean and atmosphere. In order to obtain a comprehensive dataset of cyclone activity and ocean-atmosphere interaction a field experiment was carried out in the Lofoten region during March and April 2005. Employed platforms were the Irish research vessel RV Celtic Explorer which conducted a meteorological (radiosondes, standard parameters, observations) and an oceanographic (CTD) program. The German research aircraft Falcon accomplished eight flight missions (between 4-21 March) to observe synoptic conditions with high spatial and temporal resolution. In addition 23 autonomous marine buoys were deployed in advance of the campaign in the observed area to measure drift, air-temperature and -pressure and water-temperature. In addition to the published datasets several other measurements were performed during the experiment. Corresonding datasets will be published in the near future and are available on request. Details about all used platforms and sensors and all performed measurements are listed in the fieldreport. The following datasets are available on request: ground data at RV Celtic Explorer

Implicit equal-weights particle filter

Filter degeneracy is the main obstacle for the implementation of particle filter in non-linear high-dimensional models. A new scheme, the implicit equal-weights particle filter (IEWPF), is introduced. In this scheme samples are drawn implicitly from proposal densities with a different covariance for each particle, such that all particle weights are equal by construction. We test and explore the properties of the new scheme using a 1,000-dimensional simple linear model, and the 1,000-dimensional non-linear Lorenz96 model, and compare the performance of the scheme to a Local Ensemble Kalman Filter. The experiments show that the new scheme can easily be implemented in high-dimensional systems and is never degenerate, with good convergence properties in both systems.

Abrupt transitions in the NAO control of explosive North Atlantic cyclone development

Explosive cyclones are intense extra-tropical low pressure systems featuring large deepening rates. In the Euro-Atlantic sector, they are a major source of life-threatening weather impacts due to their associated strong wind gusts, heavy precipitation and storm surges. The wintertime variability of the North Atlantic cyclonic activity is primarily modulated by the North Atlantic Oscillation (NAO). In this study, we investigate the interannual and multi-decadal variability of explosive North Atlantic cyclones using track density data from two reanalysis datasets (NCEP and ERA-40) and a control simulation of an atmosphere/ocean coupled General Circulation Model (GCM—ECHAM5/MPIOM1). The leading interannual and multi-decadal modes of variability of explosive cyclone track density are characterized by a strengthening/weakening pattern between Newfoundland and Iceland, which is mainly modulated by the NAO at both timescales. However, the NAO control of interannual cyclone variability is not stationary in time and abruptly fluctuates during periods of 20–25 years long both in NCEP and ECHAM5/MPIOM1. These transitions are accompanied by structural changes in the leading mode of explosive cyclone variability, and by decreased/enhanced baroclinicity over the sub-polar/sub-tropical North Atlantic. The influence of the ocean is apparently important for both the occurrence and persistence of such anomalous periods. In the GCM, the Atlantic Meridional Overturning Circulation appears to influence the large-scale baroclinicity and explosive cyclone development over the North Atlantic. These results permit a better understanding of explosive cyclogenesis variability at different climatic timescales and might help to improve predictions of these hazardous events.

Spatial patterns and environmental constraints on ecosystem services at a catchment scale

Improved understanding and prediction of the fundamental environmental controls on ecosystem service supply across the landscape will help to inform decisions made by policy makers and land-water managers. To evaluate this issue for a local catchment case study, we explored metrics and spatial patterns of service supply for water quality regulation, agriculture production, carbon storage, and biodiversity for the Macronutrient Conwy catchment. Methods included using ecosystem models such as LUCI and JULES, integration of national scale field survey datasets, earth observation products and plant trait databases, to produce finely resolved maps of species richness and primary production. Analyses were done with both 1x1 km gridded and subcatchment data. A common single gradient characterised catchment scale ecosystem services supply with agricultural production and carbon storage at opposing ends of the gradient as reported for a national-scale assessment. Species diversity was positively related to production due to the below national average productivity levels in the Conwy combined with the unimodal relationship between biodiversity and productivity at the national scale. In contrast to the national scale assessment, a strong reduction in water quality as production increased was observed in these low productive systems. Various soil variables were tested for their predictive power of ecosystem service supply. Soil carbon, nitrogen, their ratio and soil pH all had double the power of rainfall and altitude, each explaining around 45% of variation but soil pH is proposed as a potential metric for ecosystem service supply potential as it is a simple and practical metric which can be carried out in the field with crowd-sourcing technologies now available. The study emphasises the importance of considering multiple ecosystem services together due to the complexity of covariation at local and national scales, and the benefits of exploiting a wide range of metrics for each service to enhance data robustness.

An X-ray micro-tomography system optimised for the low-dose study of living organisms

An X-ray micro-tomography system has been designed that is dedicated to the low-dose imaging of radiation sensitive living organisms and has been used to image the early development of the first few days of plant development immediately after germination. The system is based on third-generation X-ray micro-tomography system and consists of an X-ray tube, two-dimensional X-ray detector and a mechanical sample manipulation stage. The X-ray source is a 50 kVp X-ray tube with a silver target with a filter to centre the X-ray spectrum on 22 keV.A 100 mm diameter X-ray image intensifier (XRII) is used to collect the two-dimensional projection images. The rotation tomography table incorporates a linear translation mechanism to eliminate ring artefact that is commonly associated with third-generation tomography systems' Developing maize seeds (Triticum aestivum) have been imaged using the system with a cubic voxel linear dimension of 100 mum, over a diameter of 25 mm and the root lengths and volumes measured. The X-ray dose to the plants was also assessed and found to have no effect on the plant root development. (C) 2003 Elsevier Science Ltd. All rights reserved.