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.

Gut bacteria-host metabolic interplay during conventionalisation of the mouse germfree colon

The interplay between dietary nutrients, gut microbiota and mammalian host tissues of the gastrointestinal tract is recognised as highly relevant for host health. Combined transcriptome, metabonome and microbial profiling tools were employed to analyse the dynamic responses of germfree mouse colonic mucosa to colonisation by normal mouse microbiota (conventionalisation) at different time-points during 16 days. The colonising microbiota showed a shift from early (days 1 and 2) to later colonisers (days 8 and 16). The dynamic changes in the microbial community were rapidly reflected by the urine metabolic profiles (day 1) and at later stages (day 4 onward) by the colon mucosa transcriptome and metabolic profiles. Correlations of host transcriptomes, metabolite patterns and microbiota composition revealed associations between Bacilli and Proteobacteria, and differential expression of host genes involved in energy and anabolic metabolism. Differential gene expression correlated with scyllo- and myo-inositol, glutamine, glycine and alanine levels in colonic tissues during the time span of conventionalisation. Our combined time-resolved analyses may help to expand the understanding of host-microbe molecular interactions during the microbial establishment.

A defined intestinal colonization microbiota for gnotobiotic pigs

Maximising the ability of piglets to survive exposure to pathogens is essential to reduce early piglet mortality, an important factor in efficient commercial pig production. Mortality rates can be influenced by many factors, including early colonization by microbial commensals. Here we describe the development of an intestinal microbiota, the Bristol microbiota, for use in gnotobiotic pigs and its influence on synthesis of systemic immunoglobulins. Such a microbiota will be of value in studies of the consequences of early microbial colonization on development of the intestinal immune system and subsequent susceptibility to disease. Gnotobiotic pig studies lack a well-established intestinal microbiota. The use of the Altered Schaedler Flora (ASF), a murine intestinal microbiota, to colonize the intestines of Caesarean-derived, gnotobiotic pigs prior to gut closure, resulted in unreliable colonization with most (but not all) strains of the ASF. Subsequently, a novel, simpler porcine microbiota was developed. The novel microbiota reliably colonized the length of the intestinal tract when administered to gnotobiotic piglets. No health problems were observed, and the novel microbiota induced a systemic increase in serum immunoglobulins, in particular IgA and IgM. The Bristol microbiota will be of value for highly controlled, reproducible experiments of the consequences of early microbial colonization on susceptibility to disease in neonatal piglets, and as a biomedical model for the impact of microbial colonization on development of the intestinal mucosa and immune system in neonates.

TRY: a global database of plant traits

Plant traits – the morphological, anatomical, physiological, biochemical and phenological characteristics of plants and their organs – determine how primary producers respond to environmental factors, affect other trophic levels, influence ecosystem processes and services and provide a link from species richness to ecosystem functional diversity. Trait data thus represent the raw material for a wide range of research from evolutionary biology, community and functional ecology to biogeography. Here we present the global database initiative named TRY, which has united a wide range of the plant trait research community worldwide and gained an unprecedented buy-in of trait data: so far 93 trait databases have been contributed. The data repository currently contains almost three million trait entries for 69 000 out of the world's 300 000 plant species, with a focus on 52 groups of traits characterizing the vegetative and regeneration stages of the plant life cycle, including growth, dispersal, establishment and persistence. A first data analysis shows that most plant traits are approximately log-normally distributed, with widely differing ranges of variation across traits. Most trait variation is between species (interspecific), but significant intraspecific variation is also documented, up to 40% of the overall variation. Plant functional types (PFTs), as commonly used in vegetation models, capture a substantial fraction of the observed variation – but for several traits most variation occurs within PFTs, up to 75% of the overall variation. In the context of vegetation models these traits would better be represented by state variables rather than fixed parameter values. The improved availability of plant trait data in the unified global database is expected to support a paradigm shift from species to trait-based ecology, offer new opportunities for synthetic plant trait research and enable a more realistic and empirically grounded representation of terrestrial vegetation in Earth system models.

Present state of global wetland extent and wetland methane modelling: conclusions from a model inter-comparison project (WETCHIMP)

Global wetlands are believed to be climate sensitive, and are the largest natural emitters of methane (CH4). Increased wetland CH4 emissions could act as a positive feedback to future warming. The Wetland and Wetland CH4 Inter-comparison of Models Project (WETCHIMP) investigated our present ability to simulate large-scale wetland characteristics and corresponding CH4 emissions. To ensure inter-comparability, we used a common experimental protocol driving all models with the same climate and carbon dioxide (CO2) forcing datasets. The WETCHIMP experiments were conducted for model equilibrium states as well as transient simulations covering the last century. Sensitivity experiments investigated model response to changes in selected forcing inputs (precipitation, temperature, and atmospheric CO2 concentration). Ten models participated, covering the spectrum from simple to relatively complex, including models tailored either for regional or global simulations. The models also varied in methods to calculate wetland size and location, with some models simulating wetland area prognostically, while other models relied on remotely sensed inundation datasets, or an approach intermediate between the two. Four major conclusions emerged from the project. First, the suite of models demonstrate extensive disagreement in their simulations of wetland areal extent and CH4 emissions, in both space and time. Simple metrics of wetland area, such as the latitudinal gradient, show large variability, principally between models that use inundation dataset information and those that independently determine wetland area. Agreement between the models improves for zonally summed CH4 emissions, but large variation between the models remains. For annual global CH4 emissions, the models vary by ±40% of the all-model mean (190 Tg CH4 yr−1). Second, all models show a strong positive response to increased atmospheric CO2 concentrations (857 ppm) in both CH4 emissions and wetland area. In response to increasing global temperatures (+3.4 °C globally spatially uniform), on average, the models decreased wetland area and CH4 fluxes, primarily in the tropics, but the magnitude and sign of the response varied greatly. Models were least sensitive to increased global precipitation (+3.9 % globally spatially uniform) with a consistent small positive response in CH4 fluxes and wetland area. Results from the 20th century transient simulation show that interactions between climate forcings could have strong non-linear effects. Third, we presently do not have sufficient wetland methane observation datasets adequate to evaluate model fluxes at a spatial scale comparable to model grid cells (commonly 0.5°). This limitation severely restricts our ability to model global wetland CH4 emissions with confidence. Our simulated wetland extents are also difficult to evaluate due to extensive disagreements between wetland mapping and remotely sensed inundation datasets. Fourth, the large range in predicted CH4 emission rates leads to the conclusion that there is both substantial parameter and structural uncertainty in large-scale CH4 emission models, even after uncertainties in wetland areas are accounted for.

Efficient fully nonlinear data assimilation for geophysical fluid dynamics

A potential problem with Ensemble Kalman Filter is the implicit Gaussian assumption at analysis times. Here we explore the performance of a recently proposed fully nonlinear particle filter on a high-dimensional but simplified ocean model, in which the Gaussian assumption is not made. The model simulates the evolution of the vorticity field in time, described by the barotropic vorticity equation, in a highly nonlinear flow regime. While common knowledge is that particle filters are inefficient and need large numbers of model runs to avoid degeneracy, the newly developed particle filter needs only of the order of 10-100 particles on large scale problems. The crucial new ingredient is that the proposal density cannot only be used to ensure all particles end up in high-probability regions of state space as defined by the observations, but also to ensure that most of the particles have similar weights. Using identical twin experiments we found that the ensemble mean follows the truth reliably, and the difference from the truth is captured by the ensemble spread. A rank histogram is used to show that the truth run is indistinguishable from any of the particles, showing statistical consistency of the method.

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.

Cloud banding and winds in intense European cyclones: results from the DIAMET project

The DIAMET (DIAbatic influences on Mesoscale structures in ExTratropical storms) project aims to improve forecasts of high-impact weather in extratropical cyclones through field measurements, high-resolution numerical modeling, and improved design of ensemble forecasting and data assimilation systems. This article introduces DIAMET and presents some of the first results. Four field campaigns were conducted by the project, one of which, in late 2011, coincided with an exceptionally stormy period marked by an unusually strong, zonal North Atlantic jet stream and a succession of severe windstorms in northwest Europe. As a result, December 2011 had the highest monthly North Atlantic Oscillation index (2.52) of any December in the last 60 years. Detailed observations of several of these storms were gathered using the UK’s BAe146 research aircraft and extensive ground-based measurements. As an example of the results obtained during the campaign, observations are presented of cyclone Friedhelm on 8 December 2011, when surface winds with gusts exceeding 30 m s-1 crossed central Scotland, leading to widespread disruption to transportation and electricity supply. Friedhelm deepened 44 hPa in 24 hours and developed a pronounced bent-back front wrapping around the storm center. The strongest winds at 850 hPa and the surface occurred in the southern quadrant of the storm, and detailed measurements showed these to be most intense in clear air between bands of showers. High-resolution ensemble forecasts from the Met Office showed similar features, with the strongest winds aligned in linear swaths between the bands, suggesting that there is potential for improved skill in forecasts of damaging winds.

Observation impact in data assimilation: the effect of non-Gaussian observation error.

Data assimilation methods which avoid the assumption of Gaussian error statistics are being developed for geoscience applications. We investigate how the relaxation of the Gaussian assumption affects the impact observations have within the assimilation process. The effect of non-Gaussian observation error (described by the likelihood) is compared to previously published work studying the effect of a non-Gaussian prior. The observation impact is measured in three ways: the sensitivity of the analysis to the observations, the mutual information, and the relative entropy. These three measures have all been studied in the case of Gaussian data assimilation and, in this case, have a known analytical form. It is shown that the analysis sensitivity can also be derived analytically when at least one of the prior or likelihood is Gaussian. This derivation shows an interesting asymmetry in the relationship between analysis sensitivity and analysis error covariance when the two different sources of non-Gaussian structure are considered (likelihood vs. prior). This is illustrated for a simple scalar case and used to infer the effect of the non-Gaussian structure on mutual information and relative entropy, which are more natural choices of metric in non-Gaussian data assimilation. It is concluded that approximating non-Gaussian error distributions as Gaussian can give significantly erroneous estimates of observation impact. The degree of the error depends not only on the nature of the non-Gaussian structure, but also on the metric used to measure the observation impact and the source of the non-Gaussian structure.

Gaussian anamorphosis in the analysis step of the EnKF: a joint state-variable/observation approach

The analysis step of the (ensemble) Kalman filter is optimal when (1) the distribution of the background is Gaussian, (2) state variables and observations are related via a linear operator, and (3) the observational error is of additive nature and has Gaussian distribution. When these conditions are largely violated, a pre-processing step known as Gaussian anamorphosis (GA) can be applied. The objective of this procedure is to obtain state variables and observations that better fulfil the Gaussianity conditions in some sense. In this work we analyse GA from a joint perspective, paying attention to the effects of transformations in the joint state variable/observation space. First, we study transformations for state variables and observations that are independent from each other. Then, we introduce a targeted joint transformation with the objective to obtain joint Gaussianity in the transformed space. We focus primarily in the univariate case, and briefly comment on the multivariate one. A key point of this paper is that, when (1)-(3) are violated, using the analysis step of the EnKF will not recover the exact posterior density in spite of any transformations one may perform. These transformations, however, provide approximations of different quality to the Bayesian solution of the problem. Using an example in which the Bayesian posterior can be analytically computed, we assess the quality of the analysis distributions generated after applying the EnKF analysis step in conjunction with different GA options. The value of the targeted joint transformation is particularly clear for the case when the prior is Gaussian, the marginal density for the observations is close to Gaussian, and the likelihood is a Gaussian mixture.

Recovery based on plot experiments is a poor predictor of landscape-level population impacts of agricultural pesticides

Current European Union regulatory risk assessment allows application of pesticides provided that recovery of nontarget arthropods in-crop occurs within a year. Despite the long-established theory of source-sink dynamics, risk assessment ignores depletion of surrounding populations and typical field trials are restricted to plot-scale experiments. In the present study, the authors used agent-based modeling of 2 contrasting invertebrates, a spider and a beetle, to assess how the area of pesticide application and environmental half-life affect the assessment of recovery at the plot scale and impact the population at the landscape scale. Small-scale plot experiments were simulated for pesticides with different application rates and environmental half-lives. The same pesticides were then evaluated at the landscape scale (10 km × 10 km) assuming continuous year-on-year usage. The authors' results show that recovery time estimated from plot experiments is a poor indicator of long-term population impact at the landscape level and that the spatial scale of pesticide application strongly determines population-level impact. This raises serious doubts as to the utility of plot-recovery experiments in pesticide regulatory risk assessment for population-level protection. Predictions from the model are supported by empirical evidence from a series of studies carried out in the decade starting in 1988. The issues raised then can now be addressed using simulation. Prediction of impacts at landscape scales should be more widely used in assessing the risks posed by environmental stressors.

Representation errors and retrievals in linear and nonlinear data assimilation

This article shows how one can formulate the representation problem starting from Bayes’ theorem. The purpose of this article is to raise awareness of the formal solutions,so that approximations can be placed in a proper context. The representation errors appear in the likelihood, and the different possibilities for the representation of reality in model and observations are discussed, including nonlinear representation probability density functions. Specifically, the assumptions needed in the usual procedure to add a representation error covariance to the error covariance of the observations are discussed,and it is shown that, when several sub-grid observations are present, their mean still has a representation error ; socalled ‘superobbing’ does not resolve the issue. Connection is made to the off-line or on-line retrieval problem, providing a new simple proof of the equivalence of assimilating linear retrievals and original observations. Furthermore, it is shown how nonlinear retrievals can be assimilated without loss of information. Finally we discuss how errors in the observation operator model can be treated consistently in the Bayesian framework, connecting to previous work in this area.

The AeroCom evaluation and intercomparison of organic aerosol in global models

This paper evaluates the current status of global modeling of the organic aerosol (OA) in the troposphere and analyzes the differences between models as well as between models and observations. Thirty-one global chemistry transport models (CTMs) and general circulation models (GCMs) have participated in this intercomparison, in the framework of AeroCom phase II. The simulation of OA varies greatly between models in terms of the magnitude of primary emissions, secondary OA (SOA) formation, the number of OA species used (2 to 62), the complexity of OA parameterizations (gas-particle partitioning, chemical aging, multiphase chemistry, aerosol microphysics), and the OA physical, chemical and optical properties. The diversity of the global OA simulation results has increased since earlier AeroCom experiments, mainly due to the increasing complexity of the SOA parameterization in models, and the implementation of new, highly uncertain, OA sources. Diversity of over one order of magnitude exists in the modeled vertical distribution of OA concentrations that deserves a dedicated future study. Furthermore, although the OA / OC ratio depends on OA sources and atmospheric processing, and is important for model evaluation against OA and OC observations, it is resolved only by a few global models. The median global primary OA (POA) source strength is 56 Tg a−1 (range 34–144 Tg a−1) and the median SOA source strength (natural and anthropogenic) is 19 Tg a−1 (range 13–121 Tg a−1). Among the models that take into account the semi-volatile SOA nature, the median source is calculated to be 51 Tg a−1 (range 16–121 Tg a−1), much larger than the median value of the models that calculate SOA in a more simplistic way (19 Tg a−1; range 13–20 Tg a−1, with one model at 37 Tg a−1). The median atmospheric burden of OA is 1.4 Tg (24 models in the range of 0.6–2.0 Tg and 4 between 2.0 and 3.8 Tg), with a median OA lifetime of 5.4 days (range 3.8–9.6 days). In models that reported both OA and sulfate burdens, the median value of the OA/sulfate burden ratio is calculated to be 0.77; 13 models calculate a ratio lower than 1, and 9 models higher than 1. For 26 models that reported OA deposition fluxes, the median wet removal is 70 Tg a−1 (range 28–209 Tg a−1), which is on average 85% of the total OA deposition. Fine aerosol organic carbon (OC) and OA observations from continuous monitoring networks and individual field campaigns have been used for model evaluation. At urban locations, the model–observation comparison indicates missing knowledge on anthropogenic OA sources, both strength and seasonality. The combined model–measurements analysis suggests the existence of increased OA levels during summer due to biogenic SOA formation over large areas of the USA that can be of the same order of magnitude as the POA, even at urban locations, and contribute to the measured urban seasonal pattern. Global models are able to simulate the high secondary character of OA observed in the atmosphere as a result of SOA formation and POA aging, although the amount of OA present in the atmosphere remains largely underestimated, with a mean normalized bias (MNB) equal to −0.62 (−0.51) based on the comparison against OC (OA) urban data of all models at the surface, −0.15 (+0.51) when compared with remote measurements, and −0.30 for marine locations with OC data. The mean temporal correlations across all stations are low when compared with OC (OA) measurements: 0.47 (0.52) for urban stations, 0.39 (0.37) for remote stations, and 0.25 for marine stations with OC data. The combination of high (negative) MNB and higher correlation at urban stations when compared with the low MNB and lower correlation at remote sites suggests that knowledge about the processes that govern aerosol processing, transport and removal, on top of their sources, is important at the remote stations. There is no clear change in model skill with increasing model complexity with regard to OC or OA mass concentration. However, the complexity is needed in models in order to distinguish between anthropogenic and natural OA as needed for climate mitigation, and to calculate the impact of OA on climate accurately.

Comparing hybrid data assimilation methods on the Lorenz 1963 model with increasing nonlinearity

We systematically compare the performance of ETKF-4DVAR, 4DVAR-BEN and 4DENVAR with respect to two traditional methods (4DVAR and ETKF) and an ensemble transform Kalman smoother (ETKS) on the Lorenz 1963 model. We specifically investigated this performance with increasing nonlinearity and using a quasi-static variational assimilation algorithm as a comparison. Using the analysis root mean square error (RMSE) as a metric, these methods have been compared considering (1) assimilation window length and observation interval size and (2) ensemble size to investigate the influence of hybrid background error covariance matrices and nonlinearity on the performance of the methods. For short assimilation windows with close to linear dynamics, it has been shown that all hybrid methods show an improvement in RMSE compared to the traditional methods. For long assimilation window lengths in which nonlinear dynamics are substantial, the variational framework can have diffculties fnding the global minimum of the cost function, so we explore a quasi-static variational assimilation (QSVA) framework. Of the hybrid methods, it is seen that under certain parameters, hybrid methods which do not use a climatological background error covariance do not need QSVA to perform accurately. Generally, results show that the ETKS and hybrid methods that do not use a climatological background error covariance matrix with QSVA outperform all other methods due to the full flow dependency of the background error covariance matrix which also allows for the most nonlinearity.