Statistics of spatial averages and optimal averaging in the presence of missing data

We consider statistics of spatial averages estimated by weighting observations over an arbitrary spatial domain using identical and independent measuring devices, and derive an account of bias and variance in the presence of missing observations. We test the model relative to simulations, and the approximations for bias and variance with missing data are shown to compare well even when the probability of missing data is large. Previous authors have examined optimal averaging strategies for minimizing bias, variance and mean squared error of the spatial average, and we extend the analysis to the case of missing observations. Minimizing variance mainly requires higher weights where local variance and covariance is small, whereas minimizing bias requires higher weights where the field is closer to the true spatial average. Missing data increases variance and contributes to bias, and reducing both effects involves emphasizing locations with mean value nearer to the spatial average. The framework is applied to study spatially averaged rainfall over India. We use our model to estimate standard error in all-India rainfall as the combined effect of measurement uncertainty and bias, when weights are chosen so as to yield minimum mean squared error

Convergent estimators of variance of a spatial mean in the presence of missing observations

In the geosciences, a recurring problem is one of estimating spatial means of a physical field using weighted averages of point observations. An important variant is when individual observations are counted with some probability less than one. This can occur in different contexts: from missing data to estimating the statistics across subsamples. In such situations, the spatial mean is a ratio of random variables, whose statistics involve approximate estimators derived through series expansion. The present paper considers truncated estimators of variance of the spatial mean and their general structure in the presence of missing data. To all orders, the variance estimator depends only on the first and second moments of the underlying field, and convergence requires these moments to be finite. Furthermore, convergence occurs if either the probability of counting individual observations is larger than 1/2 or the number of point observations is large. In case the point observations are weighted uniformly, the estimators are easily found using combinatorics and involve Stirling numbers of the second kind

Invariants and chaos in the Volterra gyrostat without energy conservation

The model of the Volterra gyrostat (VG) has not only played an important role in rigid body dynamics but also served as the foundation of low-order models of many naturally occurring systems. It is well known that VG possesses two invariants, or constants of motion, corresponding to kinetic energy and squared angular momentum, giving oscillatory solutions to its equations of motion. Nine distinct subclasses of the VG have been identified, two of which the Euler gyroscope and Lorenz gyrostat are each known to have two constants. This paper provides a complete characterization of constants of motion of the VG and its subclasses, showing how these enjoy two constants of motion even when rendered in terms of a non-invertible transformation of parameters, leading to a transformed Volterra gyrostat (tVG). If the quadratic coefficients of the tVG sum to zero, as they do for the VG, the system conserves energy. In all of these cases, the flows preserve volume; however, physical models where the quadratic coefficients do not sum to zero are ubiquitous, and characterization of constants of motion and the resulting dynamics for this more general class of models with volume conservation but without energy conservation is lacking. We provide such a characterization for each of the subclasses. Those with three linear feedback terms have no constants of motion, and thereby admit rich dynamics including chaos. This gives rise to a broad class of three-dimensional volume conserving chaotic flows, arising naturally from model reduction techniques

Minimal chaotic models from the Volterra gyrostat

Low-order models obtained through Galerkin projection of several physically important systems (e.g., Rayleigh-B\'enard convection, mid-latitude quasi-geostrophic dynamics, and vorticity dynamics) appear in the form of coupled gyrostats. Forced dissipative chaos is an important phenomenon in these models, and this paper considers the minimal chaotic models, in the sense of having the fewest external forcing and linear dissipation terms, arising from an underlying gyrostat core. It is shown here that a critical distinction is whether the gyrostat core (without forcing or dissipation) conserves energy, depending on whether the sum of the quadratic coefficients is zero. The paper demonstrates that, for the energy-conserving case of the gyrostat core, the requirement of a characteristic pair of fixed points that repel the chaotic flow dictates placement of forcing and dissipation in the minimal chaotic models. In contrast, if the core does not conserve energy, the forcing can be arranged in additional ways for chaos to appear, especially for the cases where linear feedbacks render fewer invariants in the gyrostat core. In all cases, the linear mode must experience dissipation for chaos to arise. Thus, the Volterra gyrostat presents a clear example where the arrangement of fixed points circumscribes more complex dynamics

Reducing the environmental impact of surgery on a global scale: systematic review and co-prioritization with healthcare workers in 132 countries

Abstract Background Healthcare cannot achieve net-zero carbon without addressing operating theatres. The aim of this study was to prioritize feasible interventions to reduce the environmental impact of operating theatres. Methods This study adopted a four-phase Delphi consensus co-prioritization methodology. In phase 1, a systematic review of published interventions and global consultation of perioperative healthcare professionals were used to longlist interventions. In phase 2, iterative thematic analysis consolidated comparable interventions into a shortlist. In phase 3, the shortlist was co-prioritized based on patient and clinician views on acceptability, feasibility, and safety. In phase 4, ranked lists of interventions were presented by their relevance to high-income countries and low–middle-income countries. Results In phase 1, 43 interventions were identified, which had low uptake in practice according to 3042 professionals globally. In phase 2, a shortlist of 15 intervention domains was generated. In phase 3, interventions were deemed acceptable for more than 90 per cent of patients except for reducing general anaesthesia (84 per cent) and re-sterilization of ‘single-use’ consumables (86 per cent). In phase 4, the top three shortlisted interventions for high-income countries were: introducing recycling; reducing use of anaesthetic gases; and appropriate clinical waste processing. In phase 4, the top three shortlisted interventions for low–middle-income countries were: introducing reusable surgical devices; reducing use of consumables; and reducing the use of general anaesthesia. Conclusion This is a step toward environmentally sustainable operating environments with actionable interventions applicable to both high– and low–middle–income countries

Economic tradeoffs in mitigation, due to different atmospheric lifetimes of CO2 and black carbon

Tradeoffs are examined between mitigating black carbon (BC) and carbon dioxide (CO2) for limiting peak global mean warming, using the following set of methods. A two-box climate model is used to simulate temperatures of the atmosphere and ocean for different rates of mitigation. Mitigation rates for BC and CO2 are characterized by respective timescales for e-folding reduction in emissions intensity of gross global product. There are respective emissions models that force the box model. Lastly there is a simple economics model, with cost of mitigation varying inversely with emission intensity. Constant mitigation timescale corresponds to mitigation at a constant annual rate, for example an e-folding timescale of 40 years corresponds to 2.5% reduction each year. Discounted present cost depends only on respective mitigation timescale and respective mitigation cost at present levels of emission intensity. Least-cost mitigation is posed as choosing respective e-folding timescales, to minimize total mitigation cost under a temperature constraint (e.g. within 2 degrees C above preindustrial). Peak warming is more sensitive to mitigation timescale for CO2 than for BC. Therefore rapid mitigation of CO2 emission intensity is essential to limiting peak warming, but simultaneous mitigation of BC can reduce total mitigation expenditure. (c) 2015 Elsevier B.V. All rights reserved

Fast-slow climate dynamics and peak global warming

The dynamics of a linear two-box energy balance climate model is analyzed as a fast-slow system, where the atmosphere, land, and near-surface ocean taken together respond within few years to external forcing whereas the deep-ocean responds much more slowly. Solutions to this system are approximated by estimating the system's time-constants using a first-order expansion of the system's eigenvalue problem in a perturbation parameter, which is the ratio of heat capacities of upper and lower boxes. The solution naturally admits an interpretation in terms of a fast response that depends approximately on radiative forcing and a slow response depending on integrals of radiative forcing with respect to time. The slow response is inversely proportional to the ``damping-timescale'', the timescale with which deep-ocean warming influences global warming. Applications of approximate solutions are discussed: conditions for a warming peak, effects of an individual pulse emission of carbon dioxide (CO), and metrics for estimating and comparing contributions of different climate forcers to maximum global warming