Nonlinear data assimilation in geosciences: an extremely efficient particle filter

Almost all research fields in geosciences use numerical models and observations and combine these using data-assimilation techniques. With ever-increasing resolution and complexity, the numerical models tend to be highly nonlinear and also observations become more complicated and their relation to the models more nonlinear. Standard data-assimilation techniques like (ensemble) Kalman filters and variational methods like 4D-Var rely on linearizations and are likely to fail in one way or another. Nonlinear data-assimilation techniques are available, but are only efficient for small-dimensional problems, hampered by the so-called ‘curse of dimensionality’. Here we present a fully nonlinear particle filter that can be applied to higher dimensional problems by exploiting the freedom of the proposal density inherent in particle filtering. The method is illustrated for the three-dimensional Lorenz model using three particles and the much more complex 40-dimensional Lorenz model using 20 particles. By also applying the method to the 1000-dimensional Lorenz model, again using only 20 particles, we demonstrate the strong scale-invariance of the method, leading to the optimistic conjecture that the method is applicable to realistic geophysical problems. Copyright c 2010 Royal Meteorological Society

State estimation using the particle filter with mode tracking

A particle filter is a data assimilation scheme that employs a fully nonlinear, non-Gaussian analysis step. Unfortunately as the size of the state grows the number of ensemble members required for the particle filter to converge to the true solution increases exponentially. To overcome this Vaswani [Vaswani N. 2008. IEEE Trans Signal Process 56:4583–97] proposed a new method known as mode tracking to improve the efficiency of the particle filter. When mode tracking, the state is split into two subspaces. One subspace is forecast using the particle filter, the other is treated so that its values are set equal to the mode of the marginal pdf. There are many ways to split the state. One hypothesis is that the best results should be obtained from the particle filter with mode tracking when we mode track the maximum number of unimodal dimensions. The aim of this paper is to test this hypothesis using the three dimensional stochastic Lorenz equations with direct observations. It is found that mode tracking the maximum number of unimodal dimensions does not always provide the best result. The best choice of states to mode track depends on the number of particles used and the accuracy and frequency of the observations.

Kinetic multi-layer model of gas-particle interactions in aerosols and clouds (KM-GAP): linking condensation, evaporation and chemical reactions of organics, oxidants and water

We present a novel kinetic multi-layer model for gas-particle interactions in aerosols and clouds (KM-GAP) that treats explicitly all steps of mass transport and chemical reaction of semi-volatile species partitioning between gas phase, particle surface and particle bulk. KM-GAP is based on the PRA model framework (Pöschl-Rudich-Ammann, 2007), and it includes gas phase diffusion, reversible adsorption, surface reactions, bulk diffusion and reaction, as well as condensation, evaporation and heat transfer. The size change of atmospheric particles and the temporal evolution and spatial profile of the concentration of individual chemical species can be modelled along with gas uptake and accommodation coefficients. Depending on the complexity of the investigated system, unlimited numbers of semi-volatile species, chemical reactions, and physical processes can be treated, and the model shall help to bridge gaps in the understanding and quantification of multiphase chemistry and microphysics in atmo- spheric aerosols and clouds. In this study we demonstrate how KM-GAP can be used to analyze, interpret and design experimental investigations of changes in particle size and chemical composition in response to condensation, evaporation, and chemical reaction. For the condensational growth of water droplets, our kinetic model results provide a direct link between laboratory observations and molecular dynamic simulations, confirming that the accommodation coefficient of water at 270 K is close to unity. Literature data on the evaporation of dioctyl phthalate as a function of particle size and time can be reproduced, and the model results suggest that changes in the experimental conditions like aerosol particle concentration and chamber geometry may influence the evaporation kinetics and can be optimized for eðcient probing of specific physical effects and parameters. With regard to oxidative aging of organic aerosol particles, we illustrate how the formation and evaporation of volatile reaction products like nonanal can cause a decrease in the size of oleic acid particles exposed to ozone.

The system identification and control of Hammerstein system using non-uniform rational B-spline neural network and particle swarm optimization

In this paper a new system identification algorithm is introduced for Hammerstein systems based on observational input/output data. The nonlinear static function in the Hammerstein system is modelled using a non-uniform rational B-spline (NURB) neural network. The proposed system identification algorithm for this NURB network based Hammerstein system consists of two successive stages. First the shaping parameters in NURB network are estimated using a particle swarm optimization (PSO) procedure. Then the remaining parameters are estimated by the method of the singular value decomposition (SVD). Numerical examples including a model based controller are utilized to demonstrate the efficacy of the proposed approach. The controller consists of computing the inverse of the nonlinear static function approximated by NURB network, followed by a linear pole assignment controller.

Kinetic multi-layer model of gas-particle interactions in aerosols and clouds (KM-GAP): linking condensation, evaporation and chemical reactions of organics, oxidants and water

We present a novel kinetic multi-layer model for gas-particle interactions in aerosols and clouds (KMGAP) that treats explicitly all steps of mass transport and chemical reaction of semi-volatile species partitioning between gas phase, particle surface and particle bulk. KMGAP is based on the PRA model framework (P¨oschl-Rudich- Ammann, 2007), and it includes gas phase diffusion, reversible adsorption, surface reactions, bulk diffusion and reaction, as well as condensation, evaporation and heat transfer. The size change of atmospheric particles and the temporal evolution and spatial profile of the concentration of individual chemical species can be modeled along with gas uptake and accommodation coefficients. Depending on the complexity of the investigated system and the computational constraints, unlimited numbers of semi-volatile species, chemical reactions, and physical processes can be treated, and the model shall help to bridge gaps in the understanding and quantification of multiphase chemistry and microphysics in atmospheric aerosols and clouds. In this study we demonstrate how KM-GAP can be used to analyze, interpret and design experimental investigations of changes in particle size and chemical composition in response to condensation, evaporation, and chemical reaction. For the condensational growth of water droplets, our kinetic model results provide a direct link between laboratory observations and molecular dynamic simulations, confirming that the accommodation coefficient of water at 270K is close to unity (Winkler et al., 2006). Literature data on the evaporation of dioctyl phthalate as a function of particle size and time can be reproduced, and the model results suggest that changes in the experimental conditions like aerosol particle concentration and chamber geometry may influence the evaporation kinetics and can be optimized for efficient probing of specific physical effects and parameters. With regard to oxidative aging of organic aerosol particles, we illustrate how the formation and evaporation of volatile reaction products like nonanal can cause a decrease in the size of oleic acid particles exposed to ozone.

Reworking the NAFTA: departures from traditional frameworks

This paper reviews the treatment of intellectual property rights in the North American Free Trade Agreement (NAFTA) and considers the welfare-theoretic bases for innovation transfer between member and nonmember states. Specifically, we consider the effects of new technology development from within the union and question whether it is efficient (in a welfare sense) to transfer that new technology to nonmember states. When the new technology contains stochastic components, the important issue of information exchange arises and we consider this question in a simple oligopoly model with Bayesian updating. In this context, it is natural to ask the optimal price at which such information should be transferred. Some simple, natural conjugate examples are used to motivate the key parameters upon which the answer is dependent

Particle swarm optimisation assisted classification using elastic net prefiltering

A novel two-stage construction algorithm for linear-in-the-parameters classifier is proposed, aiming at noisy two-class classification problems. The purpose of the first stage is to produce a prefiltered signal that is used as the desired output for the second stage to construct a sparse linear-in-the-parameters classifier. For the first stage learning of generating the prefiltered signal, a two-level algorithm is introduced to maximise the model's generalisation capability, in which an elastic net model identification algorithm using singular value decomposition is employed at the lower level while the two regularisation parameters are selected by maximising the Bayesian evidence using a particle swarm optimization algorithm. Analysis is provided to demonstrate how “Occam's razor” is embodied in this approach. The second stage of sparse classifier construction is based on an orthogonal forward regression with the D-optimality algorithm. Extensive experimental results demonstrate that the proposed approach is effective and yields competitive results for noisy data sets.

Aircraft observations and model simulations of concentration and particle size distribution in the Eyjafjallajökull volcanic ash cloud

The Eyjafjallajökull volcano in Iceland emitted a cloud of ash into the atmosphere during April and May 2010. Over the UK the ash cloud was observed by the FAAM BAe-146 Atmospheric Research Aircraft which was equipped with in-situ probes measuring the concentration of volcanic ash carried by particles of varying sizes. The UK Met Office Numerical Atmospheric-dispersion Modelling Environment (NAME) has been used to simulate the evolution of the ash cloud emitted by the Eyjafjallajökull volcano during the period 4–18 May 2010. In the NAME simulations the processes controlling the evolution of the concentration and particle size distribution include sedimentation and deposition of particles, horizontal dispersion and vertical wind shear. For travel times between 24 and 72 h, a 1/t relationship describes the evolution of the concentration at the centre of the ash cloud and the particle size distribution remains fairly constant. Although NAME does not represent the effects of microphysical processes, it can capture the observed decrease in concentration with travel time in this period. This suggests that, for this eruption, microphysical processes play a small role in determining the evolution of the distal ash cloud. Quantitative comparison with observations shows that NAME can simulate the observed column-integrated mass if around 4% of the total emitted mass is assumed to be transported as far as the UK by small particles (< 30 μm diameter). NAME can also simulate the observed particle size distribution if a distal particle size distribution that contains a large fraction of < 10 μm diameter particles is used, consistent with the idea that phraetomagmatic volcanoes, such as Eyjafjallajökull, emit very fine particles.

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.

Sensitivity of direct radiative forcing by mineral dust to particle characteristics

Airborne dust affects the Earth's energy balance — an impact that is measured in terms of the implied change in net radiation (or radiative forcing, in W m-2) at the top of the atmosphere. There remains considerable uncertainty in the magnitude and sign of direct forcing by airborne dust under current climate. Much of this uncertainty stems from simplified assumptions about mineral dust-particle size, composition and shape, which are applied in remote sensing retrievals of dust characteristics and dust-cycle models. Improved estimates of direct radiative forcing by dust will require improved characterization of the spatial variability in particle characteristics to provide reliable information dust optical properties. This includes constraints on: (1) particle-size distribution, including discrimination of particle subpopulations and quantification of the amount of dust in the sub-10 µm to <0.1 µm mass fraction; (2) particle composition, specifically the abundance of iron oxides, and whether particles consist of single or multi-mineral grains; (3) particle shape, including degree of sphericity and surface roughness, as a function of size and mineralogy; and (4) the degree to which dust particles are aggregated together. The use of techniques that measure the size, composition and shape of individual particles will provide a better basis for optical modelling.

Self-consistent wave-particle interactions in dispersive scale long-period field-line-resonances

Using 1D Vlasov drift-kinetic computer simulations, it is shown that electron trapping in long period standing shear Alfven waves (SAWs) provides an efficient energy sink for wave energy that is much more effective than Landau damping. It is also suggested that the plasma environment of low altitude auroral-zone geomagnetic field lines is more suited to electron acceleration by inertial or kinetic scale Alfven waves. This is due to the self-consistent response of the electron distribution function to SAWs, which must accommodate the low altitude large-scale current system in standing waves. We characterize these effects in terms of the relative magnitude of the wave phase and electron thermal velocities. While particle trapping is shown to be significant across a wide range of plasma temperatures and wave frequencies, we find that electron beam formation in long period waves is more effective in relatively cold plasma.

High-latitude particle precipitation and its relationship to magnetospheric source regions

Two central issues in magnetospheric research are understanding the mapping of the low-altitude ionosphere to the distant regions of the magnetsphere, and understanding the relationship between the small-scale features detected in the various regions of the ionosphere and the global properties of the magnetosphere. The high-latitude ionosphere, through its magnetic connection to the outer magnetosphere, provides an important view of magnetospheric boundaries and the physical processes occurring there. All physical manifestations of this magnetic connectivity (waves, particle precipitation, etc.), however, have non-zero propagation times during which they are convected by the large-scale magnetospheric electric field, with phenomena undergoing different convection distances depending on their propagation times. Identification of the ionospheric signatures of magnetospheric regions and phenomena, therefore, can be difficult. Considerable progress has recently been made in identifying these convection signatures in data from low- and high-altitude satellites. This work has allowed us to learn much about issues such as: the rates of magnetic reconnection, both at the dayside magnetopause and in the magnetotail; particle transport across the open magnetopause; and particle acceleration at the magnetopause and the magnetotail current sheets.

Low and middle altitude cusp particle signatures for general magnetopause reconnection rate variations: 1. Theory

We present predictions of the signatures of magnetosheath particle precipitation (in the regions classified as open low-latitude boundary layer, cusp, mantle and polar cap) for periods when the interplanetary magnetic field has a southward component. These are made using the “pulsating cusp” model of the effects of time-varying magnetic reconnection at the dayside magnetopause. Predictions are made for both low-altitude satellites in the topside ionosphere and for midaltitude spacecraft in the magnetosphere. Low-altitude cusp signatures, which show a continuous ion dispersion signature, reveal "quasi-steady reconnection" (one limit of the pulsating cusp model), which persists for a period of at least 10 min. We estimate that “quasi-steady” in this context corresponds to fluctuations in the reconnection rate of a factor of 2 or less. The other limit of the pulsating cusp model explains the instantaneous jumps in the precipitating ion spectrum that have been observed at low altitudes. Such jumps are produced by isolated pulses of reconnection: that is, they are separated by intervals when the reconnection rate is zero. These also generate convecting patches on the magnetopause in which the field lines thread the boundary via a rotational discontinuity separated by more extensive regions of tangential discontinuity. Predictions of the corresponding ion precipitation signatures seen by midaltitude spacecraft are presented. We resolve the apparent contradiction between estimates of the width of the injection region from midaltitude data and the concept of continuous entry of solar wind plasma along open field lines. In addition, we reevaluate the use of pitch angle-energy dispersion to estimate the injection distance.