The propagation of errors in long-term measurements of land-atmosphere fluxes of carbon and water

For surface fluxes of carbon dioxide, the net daily flux is the sum of daytime and nighttime fluxes of approximately the same magnitude and opposite direction. The net flux is therefore significantly smaller than the individual flux measurements and error assessment is critical in determining whether a surface is a net source or sink of carbon dioxide. For carbon dioxide flux measurements, it is an occasional misconception that the net flux is measured as the difference between the net upward and downward fluxes (i.e. a small difference between large terms). This is not the case. The net flux is the sum of individual (half-hourly or hourly) flux measurements, each with an associated error term. The question of errors and uncertainties in long-term flux measurements of carbon and water is addressed by first considering the potential for errors in flux measuring systems in general and thus errors which are relevant to a wide range of timescales of measurement. We also focus exclusively on flux measurements made by the micrometeorological method of eddy covariance. Errors can loosely be divided into random errors and systematic errors, although in reality any particular error may be a combination of both types. Systematic errors can be fully systematic errors (errors that apply on all of the daily cycle) or selectively systematic errors (errors that apply to only part of the daily cycle), which have very different effects. Random errors may also be full or selective, but these do not differ substantially in their properties. We describe an error analysis in which these three different types of error are applied to a long-term dataset to discover how errors may propagate through long-term data and which can be used to estimate the range of uncertainty in the reported sink strength of the particular ecosystem studied.     

Use of the score test as a goodness-of-fit measure of the covariance structure in genetic analysis of longitudinal data

Model selection is an essential issue in longitudinal data analysis since many different models have been proposed to fit the covariance structure. The likelihood criterion is commonly used and allows to compare the fit of alternative models. Its value does not reflect, however, the potential improvement that can still be reached in fitting the data unless a reference model with the actual covariance structure is available. The score test approach does not require the knowledge of a reference model, and the score statistic has a meaningful interpretation in itself as a goodness-of-fit measure. The aim of this paper was to show how the score statistic may be separated into the genetic and environmental parts, which is difficult with the likelihood criterion, and how it can be used to check parametric assumptions made on variance and correlation parameters. Selection of models for genetic analysis was applied to a dairy cattle example for milk production.     

Modelling the Coexistence of Annual and Perennial Plants in Temporally Varying Environments*

Abstract The long-term growth and coexistence of species with large mixed populations in varying environments were modelled for representative environments and life-history characteristics of annual and perennial plants. The effects of the relationships between the means, variances, and covariances of seed yield, establishment, and survival, were explored by Taylor's expansion. The main findings are: 1. Individual variation in reproductive success within generations has no effect on long-term growth, which is determined only by the mean growth rate of the individuals of the species. 2. In annual species with nonoverlapping generations and without seed banks, the species with the largest mean log of the annual growth rate Y, that is the product of the average seed yield per plant and the establishment probability per seed, will win in competition with other species, independent of the correlations between the growth rates of the different species. In this case there is a negative tradeoff between the mean and the variance. 3. In perennial species with a lottery type of equal access to vacant sites, a high annual survival probability allows stable coexistence between perennial species with independent or negatively correlated variance in their mean annual product of seed production and establishment Y. 4. The coexistence range and the likely number of coexisting perennial species increase as a function of the variance of the common species, and is decreased by the variance of the rare species. The coexistence range is decreased by the covariance between the growth rates of the species, and between the survival of the rare species and its growth rate. 5. If mortality in the community of long-lived perennials is synchronized, the generations become nonoverlapping, and the competitive dynamics become similar to that of annuals. 6. Coexistence between annual and perennial species is promoted if the covariances between the annual survival and the relative yield of perennials, and between the yields of perennials and annuals, decrease and become more negative. 7. Selection for seed yield and establishment in different conditions in annuals favours a generalist strategy with low variance between years which provides a moderate yield and establishment over a wide range of environmental conditions. In perennial plants, long-term growth rate is determined by the lifetime seed yield and establishment. Because of strong competition with annuals in the more common conditions, selection in perennials favours instead a specialist strategy, with a high seed yield and establishment at relatively rare occasions in space and time, in which there is only weak competition with annuals. 8. Coexistence of annual species with a long-lived seed bank in the soil is also made possible by independent variation in different years of the germination, seed yield and establishment of different species, analogous to the situation of perennial plants.     

Inversion of terrestrial ecosystem model parameter values against eddy covariance measurements by Monte Carlo sampling

Effective measures to counter the rising levels of carbon dioxide in the Earth's atmosphere require that we better understand the functioning of the global carbon cycle. Uncertainties about, in particular, the terrestrial carbon cycle's response to climate change remain high. We use a well‐known stochastic inversion technique originally developed in nuclear physics, the Metropolis algorithm, to determine the full probability density functions (PDFs) of parameters of a terrestrial ecosystem model. By thus assimilating half‐hourly eddy covariance measurements of CO₂ and water fluxes, we can substantially reduce the uncertainty of approximately five model parameters, depending on prior uncertainties. Further analysis of the posterior PDF shows that almost all parameters are nearly Gaussian distributed, and reveals some distinct groups of parameters that are constrained together. We show that after assimilating only 7 days of measurements, uncertainties for net carbon uptake over 2 years for the forest site can be substantially reduced, with the median estimate in excellent agreement with measurements.