Particle filtering in geophysical systems

The application of particle filters in geophysical systems is reviewed. Some background on Bayesian filtering is provided, and the existing methods are discussed. The emphasis is on the methodology, and not so much on the applications themselves. It is shown that direct application of the basic particle filter (i.e., importance sampling using the prior as the importance density) does not work in high-dimensional systems, but several variants are shown to have potential. Approximations to the full problem that try to keep some aspects of the particle filter beyond the Gaussian approximation are also presented and discussed.

Integrating control systems defined on the frame bundles of the space forms

This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). 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. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, 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 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.

Low-order dynamical behavior in the martian atmosphere: Diagnosis of general circulation model results

The hypothesis of a low dimensional martian climate attractor is investigated by the application of the proper orthogonal decomposition (POD) to a simulation of martian atmospheric circulation using the UK Mars general circulation model (UK-MGCM). In this article we focus on a time series of the interval between autumn and winter in the northern hemisphere, when baroclinic activity is intense. The POD is a statistical technique that allows the attribution of total energy (TE) to particular structures embedded in the UK-MGCM time-evolving circulation. These structures are called empirical orthogonal functions (EOFs). Ordering the EOFs according to their associated energy content, we were able to determine the necessary number to account for a chosen amount of atmospheric TE. We show that for Mars a large fraction of TE is explained by just a few EOFs (with 90% TE in 23 EOFs), which apparently support the initial hypothesis. We also show that the resulting EOFs represent classical types of atmospheric motion, such as thermal tides and transient waves. Thus, POD is shown to be an efficient method for the identification of different classes of atmospheric modes. It also provides insight into the non-linear interaction of these modes.

Extreme value distribution for singular measures

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems that have a singular measure. Using the block maxima approach described in Faranda et al. [2011] we show that, numerically, the Extreme Value distribution for these maps can be associated to the Generalised Extreme Value family where the parameters scale with the information dimension. The numerical analysis are performed on a few low dimensional maps. For the middle third Cantor set and the Sierpinskij triangle obtained using Iterated Function Systems, experimental parameters show a very good agreement with the theoretical values. For strange attractors like Lozi and H\`enon maps a slower convergence to the Generalised Extreme Value distribution is observed. Even in presence of large statistics the observed convergence is slower if compared with the maps which have an absolute continuous invariant measure. Nevertheless and within the uncertainty computed range, the results are in good agreement with the theoretical estimates.

Nonlinear error dynamics for cycled data assimilation methods

We investigate the error dynamics for cycled data assimilation systems, such that the inverse problem of state determination is solved at tk, k = 1, 2, 3, ..., with a first guess given by the state propagated via a dynamical system model from time tk − 1 to time tk. In particular, for nonlinear dynamical systems that are Lipschitz continuous with respect to their initial states, we provide deterministic estimates for the development of the error ||ek|| := ||x(a)k − x(t)k|| between the estimated state x(a) and the true state x(t) over time. Clearly, observation error of size δ > 0 leads to an estimation error in every assimilation step. These errors can accumulate, if they are not (a) controlled in the reconstruction and (b) damped by the dynamical system under consideration. A data assimilation method is called stable, if the error in the estimate is bounded in time by some constant C. The key task of this work is to provide estimates for the error ||ek||, depending on the size δ of the observation error, the reconstruction operator Rα, the observation operator H and the Lipschitz constants K(1) and K(2) on the lower and higher modes of controlling the damping behaviour of the dynamics. We show that systems can be stabilized by choosing α sufficiently small, but the bound C will then depend on the data error δ in the form c||Rα||δ with some constant c. Since ||Rα|| → ∞ for α → 0, the constant might be large. Numerical examples for this behaviour in the nonlinear case are provided using a (low-dimensional) Lorenz '63 system.

Improving transmission reliability of low-power medium access control protocols using average diversity combining

Embedded computer systems equipped with wireless communication transceivers are nowadays used in a vast number of application scenarios. Energy consumption is important in many of these scenarios, as systems are battery operated and long maintenance-free operation is required. To achieve this goal, embedded systems employ low-power communication transceivers and protocols. However, currently used protocols cannot operate efficiently when communication channels are highly erroneous. In this study, we show how average diversity combining (ADC) can be used in state-of-the-art low-power communication protocols. This novel approach improves transmission reliability and in consequence energy consumption and transmission latency in the presence of erroneous channels. Using a testbed, we show that highly erroneous channels are indeed a common occurrence in situations, where low-power systems are used and we demonstrate that ADC improves low-power communication dramatically.

Towards a general theory of extremes for observables of chaotic dynamical systems

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the cho- sen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan–Yorke dimension of the attractor. Preliminary numer- ical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

The equivalent-weights particle filter in a high-dimensional system

In general, particle filters need large numbers of model runs in order to avoid filter degeneracy in high-dimensional systems. The recently proposed, fully nonlinear equivalent-weights particle filter overcomes this requirement by replacing the standard model transition density with two different proposal transition densities. The first proposal density is used to relax all particles towards the high-probability regions of state space as defined by the observations. The crucial second proposal density is then used to ensure that the majority of particles have equivalent weights at observation time. Here, the performance of the scheme in a high, 65 500 dimensional, simplified ocean model is explored. The success of the equivalent-weights particle filter in matching the true model state is shown using the mean of just 32 particles in twin experiments. It is of particular significance that this remains true even as the number and spatial variability of the observations are changed. The results from rank histograms are less easy to interpret and can be influenced considerably by the parameter values used. This article also explores the sensitivity of the performance of the scheme to the chosen parameter values and the effect of using different model error parameters in the truth compared with the ensemble model runs.

Aspects of particle filtering in high-dimensional spaces

Nonlinear data assimilation is high on the agenda in all fields of the geosciences as with ever increasing model resolution and inclusion of more physical (biological etc.) processes, and more complex observation operators the data-assimilation problem becomes more and more nonlinear. The suitability of particle filters to solve the nonlinear data assimilation problem in high-dimensional geophysical problems will be discussed. Several existing and new schemes will be presented and it is shown that at least one of them, the Equivalent-Weights Particle Filter, does indeed beat the curse of dimensionality and provides a way forward to solve the problem of nonlinear data assimilation in high-dimensional systems.

Solvothermal syntheses, crystal structures and properties of five new thloantimonates(III) containing the [Sb4S7](2-) anion

Five new thioantimonates have been synthesized in the presence of organic amines under solvothermal conditions and their structures determined by single-crystal X-ray diffraction. All of the compounds are layered and contain antimony-sulphide anions of stoichiometry [Sb4S7](2-), but the structure of the anion formed is dependent on the amine used in synthesis. (H3N(CH2)(4)NH3)[Sb4S7] (1) contains [Sb4S7](2-) double chains directed along [010]. Weak interchain Sb-S interactions between neighbouring chains cause the double chains to pack into layers in the ab plane. In the [001] direction, the layers of double chains alternate with doubly protonated diaminobutane molecules to which the chains are hydrogen bonded. Compounds of general formula (TH)(2)[Sb4S7] (T= CH3(CH2)(2)NH2 (2), (CH3)(2)CHNH2 (3), CH3(CH2)(3)NH2 (4) and CH3(CH2)(4)NH2 (5)) adopt a more complex structure in which [Sb3S8](7-) units are linked by Sb-3(3-) pyramids to form chains, which in turn are bridged by sulphur atoms to create sheets containing large heterorings. Pairs of such sheets form double layers of four atoms thickness that are stacked along [001]. Protonated amine molecules are located between anionic antimony-sulphide layers to which they are hydrogen bonded. Thermal analysis reveals that the decomposition temperature of materials containing [Sb4S7](2-) anions is dependent both on the structure of the anion, the lowest decomposition temperature being that of the low-dimensional phase (1) and on the identity of the amine, the decomposition temperature decreasing with an increasing number of carbon atoms and decreasing density. (c) 2005 Elsevier Inc. All rights reserved.

Different supramolecular hydrogen bond structures and significant changes in magnetic properties in dinuclear mu(2)-1,1-N-3 copper(II) complexes with very similar tridentate Schiff base blocking ligands

Three new basal-apical, mu(2)-1,1-azide bridged complexes, [CuL1(N-3)](2) (1), [CuL2(N-3)](2) (2) and [CuL3(N-3)]2 (3) with very similar tridentate Schiff base blocking ligands [L-1=N-(3-aminopropyl) salicylaldimine, L-2=7-amino-4-methyl-5-azahept-3-en-2-one and L-3=8-amino-4-methyl-5-azaoct-3-en-2-one) have been synthesised and their molecular structures determined by X-ray crystallography. In complex 1, there is no inter-dimer H-bonding. However, complexes 2 and 3 form two different supramolecular structures in which the dinuclear entities are linked by strong H-bonds giving one-dimensional systems. Variable-temperature (300-2 K) magnetic susceptibility measurements and magnetization measurements at 2 K reveal that complexes 1 and 2 have antiferromagnetic coupling while 3 has ferromagnetic coupling which is also confirmed by EPR spectra at 4-300 K. Magnetostructural correlations have been made taking into consideration both the azido bridging ligands and the existence of intermolecular hydrogen bonds in complexes 2 and 3.

Supramolecular hydrogen-bond structures and magnetic interactions in basal-apical, dinuclear, azide-bridged copper(II) complexes

The synthesis, characterisation, X-ray single crystal structures and magnetic properties of three new basal-apical mu(2)-1,1-azide-bridged complexes [(CuLN3)-N-1](2) (1), [(CuLN3)-N-2](2) (2) and [(CuLN3)-N-3](2) (3) with very similar tridentate Schiff-base blocking ligands {HL1 = N-[2-(ethylamino) ethyl] salicylaldimine; HL2 = 7-(ethylamino)-4-methyl-5-azahept-3-en-2-one; HL3 = 7-amino-4-methyl-5-azaoct-3-en-2-one} have been reported [complex 1: monoclinic, P2(1)/c, a = 8.390(2), b = 7.512(2), c = 19.822(6) Angstrom, beta = 91.45(5)degrees; complex 2: monoclinic, P2(1)/c, a = 8.070(9), b = 9.787(12), c = 15.743(17) A, beta = 98.467(10)degrees; complex 3: monoclinic, P2(1)/n, a = 5.884(7), b = 16.147(18), c = 11.901(12) Angstrom, beta = 90.050(10)degrees]. The structures consist of neutral dinuclear entities resulting from the pairing of two mononuclear units through end-on azide bridges connecting an equatorial position of one copper centre to an axial position of the other, The copper ions adopt a (4+1) square-based geometry in all the complexes. In complex 2, there is no inter-dimer hydrogen-bonding. However, complexes 1 and 3 form two different supramolecular structures in which the dinuclear entities are linked by H-bonds giving one-dimensional systems. Variable temperature (300-2 K) magnetic-susceptibility measurements and magnetisation measurements at 2 K reveal that all three complexes have antiferromagnetic coupling. Magneto-structural correlations have been made taking into consideration both the azido bridging ligands and the existence of intermolecular hydrogen bonds. ((C) Wiley-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2004).

Using the Cayley-Hamilton theorem with N-partitioned matrices

A definition is given for the characteristic equation of anN-partitioned matrix. It is then proved that this matrix satisfies its own characteristic equation. This can then be regarded as a version of the Cayley-Hamilton theorem, of use withN-dimensional systems.

Lightning-induced intensification of the ionospheric sporadic E layer

A connection between thunderstorms and the ionosphere has been hypothesized since the mid-1920s(1). Several mechanisms have been proposed to explain this connection(2-7), and evidence from modelling(8) as well as various types of measurements(9-14) demonstrate that lightning can interact with the lower ionosphere. It has been proposed, on the basis of a few observed events(15), that the ionospheric 'sporadic E' layer - transient, localized patches of relatively high electron density in the mid-ionosphere E layer, which significantly affect radio-wave propagation - can be modulated by thunderstorms, but a more formal statistical analysis is still needed. Here we identify a statistically significant intensification and descent in altitude of the mid-latitude sporadic E layer directly above thunderstorms. Because no ionospheric response to low-pressure systems without lightning is detected, we conclude that this localized intensification of the sporadic E layer can be attributed to lightning. We suggest that the co-location of lightning and ionospheric enhancement can be explained by either vertically propagating gravity waves that transfer energy from the site of lightning into the ionosphere, or vertical electrical discharge, or by a combination of these two mechanisms.