Nonlinear symmetric stability of planetary atmospheres

The energy–Casimir method is applied to the problem of symmetric stability in the context of a compressible, hydrostatic planetary atmosphere with a general equation of state. Formal stability criteria for symmetric disturbances to a zonally symmetric baroclinic flow are obtained. In the special case of a perfect gas the results of Stevens (1983) are recovered. Finite-amplitude stability conditions are also obtained that provide an upper bound on a certain positive-definite measure of disturbance amplitude.

Comparative evaluation of a new effective population size estimator based on approximate Bayesian computation

We describe and evaluate a new estimator of the effective population size (N-e), a critical parameter in evolutionary and conservation biology. This new "SummStat" N-e. estimator is based upon the use of summary statistics in an approximate Bayesian computation framework to infer N-e. Simulations of a Wright-Fisher population with known N-e show that the SummStat estimator is useful across a realistic range of individuals and loci sampled, generations between samples, and N-e values. We also address the paucity of information about the relative performance of N-e estimators by comparing the SUMMStat estimator to two recently developed likelihood-based estimators and a traditional moment-based estimator. The SummStat estimator is the least biased of the four estimators compared. In 32 of 36 parameter combinations investigated rising initial allele frequencies drawn from a Dirichlet distribution, it has the lowest bias. The relative mean square error (RMSE) of the SummStat estimator was generally intermediate to the others. All of the estimators had RMSE > 1 when small samples (n = 20, five loci) were collected a generation apart. In contrast, when samples were separated by three or more generations and Ne less than or equal to 50, the SummStat and likelihood-based estimators all had greatly reduced RMSE. Under the conditions simulated, SummStat confidence intervals were more conservative than the likelihood-based estimators and more likely to include true N-e. The greatest strength of the SummStat estimator is its flexible structure. This flexibility allows it to incorporate any, potentially informative summary statistic from Population genetic data.

A new solution of the rendering equation with stratified Monte Carlo approach

This paper is turned to the advanced Monte Carlo methods for realistic image creation. It offers a new stratified approach for solving the rendering equation. We consider the numerical solution of the rendering equation by separation of integration domain. The hemispherical integration domain is symmetrically separated into 16 parts. First 9 sub-domains are equal size of orthogonal spherical triangles. They are symmetric each to other and grouped with a common vertex around the normal vector to the surface. The hemispherical integration domain is completed with more 8 sub-domains of equal size spherical quadrangles, also symmetric each to other. All sub-domains have fixed vertices and computable parameters. The bijections of unit square into an orthogonal spherical triangle and into a spherical quadrangle are derived and used to generate sampling points. Then, the symmetric sampling scheme is applied to generate the sampling points distributed over the hemispherical integration domain. The necessary transformations are made and the stratified Monte Carlo estimator is presented. The rate of convergence is obtained and one can see that the algorithm is of super-convergent type.

A generalised Bayesian instrumental variable approach under student t-distributed errors with application

Bayesian analysis is given of an instrumental variable model that allows for heteroscedasticity in both the structural equation and the instrument equation. Specifically, the approach for dealing with heteroscedastic errors in Geweke (1993) is extended to the Bayesian instrumental variable estimator outlined in Rossi et al. (2005). Heteroscedasticity is treated by modelling the variance for each error using a hierarchical prior that is Gamma distributed. The computation is carried out by using a Markov chain Monte Carlo sampling algorithm with an augmented draw for the heteroscedastic case. An example using real data illustrates the approach and shows that ignoring heteroscedasticity in the instrument equation when it exists may lead to biased estimates.

Bayesian model comparison with un-normalised likelihoods

Models for which the likelihood function can be evaluated only up to a parameter-dependent unknown normalizing constant, such as Markov random field models, are used widely in computer science, statistical physics, spatial statistics, and network analysis. However, Bayesian analysis of these models using standard Monte Carlo methods is not possible due to the intractability of their likelihood functions. Several methods that permit exact, or close to exact, simulation from the posterior distribution have recently been developed. However, estimating the evidence and Bayes’ factors for these models remains challenging in general. This paper describes new random weight importance sampling and sequential Monte Carlo methods for estimating BFs that use simulation to circumvent the evaluation of the intractable likelihood, and compares them to existing methods. In some cases we observe an advantage in the use of biased weight estimates. An initial investigation into the theoretical and empirical properties of this class of methods is presented. Some support for the use of biased estimates is presented, but we advocate caution in the use of such estimates.

Variability of the Atlantic thermohaline circulation described by three-dimensional empirical orthogonal functions

We describe the use of bivariate 3d empirical orthogonal functions (EOFs) in characterising low frequency variability of the Atlantic thermohaline circulation (THC) in the Hadley Centre global climate model, HadCM3. We find that the leading two modes are well correlated with an index of the meridional overturning circulation (MOC) on decadal timescales, with the leading mode alone accounting for 54% of the decadal variance. Episodes of coherent oscillations in the sub-space of the leading EOFs are identified; these episodes are of great interest for the predictability of the THC, and could indicate the existence of different regimes of natural variability. The mechanism identified for the multi-decadal variability is an internal ocean mode, dominated by changes in convection in the Nordic Seas, which lead the changes in the MOC by a few years. Variations in salinity transports from the Arctic and from the North Atlantic are the main feedbacks which control the oscillation. This mode has a weak feedback onto the atmosphere and hence a surface climatic influence. Interestingly, some of these climate impacts lead the changes in the overturning. There are also similarities to observed multi-decadal climate variability.

National and sub-national assessments of soil organic carbon stocks and changes: The GEFSOC modelling system

Soil organic carbon (SOC) plays a vital role in ecosystem function, determining soil fertility, water holding capacity and susceptibility to land degradation. In addition, SOC is related to atmospheric CO, levels with soils having the potential for C release or sequestration, depending on land use, land management and climate. The United Nations Convention on Climate Change and its Kyoto Protocol, and other United Nations Conventions to Combat Desertification and on Biodiversity all recognize the importance of SOC and point to the need for quantification of SOC stocks and changes. An understanding of SOC stocks and changes at the national and regional scale is necessary to further our understanding of the global C cycle, to assess the responses of terrestrial ecosystems to climate change and to aid policy makers in making land use/management decisions. Several studies have considered SOC stocks at the plot scale, but these are site specific and of limited value in making inferences about larger areas. Some studies have used empirical methods to estimate SOC stocks and changes at the regional scale, but such studies are limited in their ability to project future changes, and most have been carried out using temperate data sets. The computational method outlined by the Intergovernmental Panel on Climate Change (IPCC) has been used to estimate SOC stock changes at the regional scale in several studies, including a recent study considering five contrasting eco regions. This 'one step' approach fails to account for the dynamic manner in which SOC changes are likely to occur following changes in land use and land management. A dynamic modelling approach allows estimates to be made in a manner that accounts for the underlying processes leading to SOC change. Ecosystem models, designed for site scale applications can be linked to spatial databases, giving spatially explicit results that allow geographic areas of change in SOC stocks to be identified. Some studies have used variations on this approach to estimate SOC stock changes at the sub-national and national scale for areas of the USA and Europe and at the watershed scale for areas of Mexico and Cuba. However, a need remained for a national and regional scale, spatially explicit system that is generically applicable and can be applied to as wide a range of soil types, climates and land uses as possible. The Global Environment Facility Soil Organic Carbon (GEFSOC) Modelling System was developed in response to this need. The GEFSOC system allows estimates of SOC stocks and changes to be made for diverse conditions, providing essential information for countries wishing to take part in an emerging C market, and bringing us closer to an understanding of the future role of soils in the global C cycle. (C) 2007 Elsevier B.V. All rights reserved.

Mesozoic climates: General circulation models and the rock record

General circulation models (GCMs) use the laws of physics and an understanding of past geography to simulate climatic responses. They are objective in character. However, they tend to require powerful computers to handle vast numbers of calculations. Nevertheless, it is now possible to compare results from different GCMs for a range of times and over a wide range of parameterisations for the past, present and future (e.g. in terms of predictions of surface air temperature, surface moisture, precipitation, etc.). GCMs are currently producing simulated climate predictions for the Mesozoic, which compare favourably with the distributions of climatically sensitive facies (e.g. coals, evaporites and palaeosols). They can be used effectively in the prediction of oceanic upwelling sites and the distribution of petroleum source rocks and phosphorites. Models also produce evaluations of other parameters that do not leave a geological record (e.g. cloud cover, snow cover) and equivocal phenomena such as storminess. Parameterisation of sub-grid scale processes is the main weakness in GCMs (e.g. land surfaces, convection, cloud behaviour) and model output for continental interiors is still too cold in winter by comparison with palaeontological data. The sedimentary and palaeontological record provides an important way that GCMs may themselves be evaluated and this is important because the same GCMs are being used currently to predict possible changes in future climate. The Mesozoic Earth was, by comparison with the present, an alien world, as we illustrate here by reference to late Triassic, late Jurassic and late Cretaceous simulations. Dense forests grew close to both poles but experienced months-long daylight in warm summers and months-long darkness in cold snowy winters. Ocean depths were warm (8 degrees C or more to the ocean floor) and reefs, with corals, grew 10 degrees of latitude further north and south than at the present time. The whole Earth was warmer than now by 6 degrees C or more, giving more atmospheric humidity and a greatly enhanced hydrological cycle. Much of the rainfall was predominantly convective in character, often focused over the oceans and leaving major desert expanses on the continental areas. Polar ice sheets are unlikely to have been present because of the high summer temperatures achieved. The model indicates extensive sea ice in the nearly enclosed Arctic seaway through a large portion of the year during the late Cretaceous, and the possibility of sea ice in adjacent parts of the Midwest Seaway over North America. The Triassic world was a predominantly warm world, the model output for evaporation and precipitation conforming well with the known distributions of evaporites, calcretes and other climatically sensitive facies for that time. The message from the geological record is clear. Through the Phanerozoic, Earth's climate has changed significantly, both on a variety of time scales and over a range of climatic states, usually baldly referred to as "greenhouse" and "icehouse", although these terms disguise more subtle states between these extremes. Any notion that the climate can remain constant for the convenience of one species of anthropoid is a delusion (although the recent rate of climatic change is exceptional). (c) 2006 Elsevier B.V. All rights reserved.

Empirical Orthogonal Functions: The Medium is the Message

Empirical orthogonal function (EOF) analysis is a powerful tool for data compression and dimensionality reduction used broadly in meteorology and oceanography. Often in the literature, EOF modes are interpreted individually, independent of other modes. In fact, it can be shown that no such attribution can generally be made. This review demonstrates that in general individual EOF modes (i) will not correspond to individual dynamical modes, (ii) will not correspond to individual kinematic degrees of freedom, (iii) will not be statistically independent of other EOF modes, and (iv) will be strongly influenced by the nonlocal requirement that modes maximize variance over the entire domain. The goal of this review is not to argue against the use of EOF analysis in meteorology and oceanography; rather, it is to demonstrate the care that must be taken in the interpretation of individual modes in order to distinguish the medium from the message.

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.

The Klein-Gordon equation in a domain with time-dependent boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time-dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.

Some general comparative points on Chao’s and Zelterman’s estimators of population size

Two simple and frequently used capture–recapture estimates of the population size are compared: Chao's lower-bound estimate and Zelterman's estimate allowing for contaminated distributions. In the Poisson case it is shown that if there are only counts of ones and twos, the estimator of Zelterman is always bounded above by Chao's estimator. If counts larger than two exist, the estimator of Zelterman is becoming larger than that of Chao's, if only the ratio of the frequencies of counts of twos and ones is small enough. A similar analysis is provided for the binomial case. For a two-component mixture of Poisson distributions the asymptotic bias of both estimators is derived and it is shown that the Zelterman estimator can experience large overestimation bias. A modified Zelterman estimator is suggested and also the bias-corrected version of Chao's estimator is considered. All four estimators are compared in a simulation study.