Analytical precision is vital in the interpretation of stable isotope data collected by secondary ion mass spectrometry (SIMS) given the small analysis volumes and the small magnitude of natural isotopic variations. The observed precision of a set of measurements is represented by the standard deviation (precision of an individual measurement) or the standard error of the mean (precision of the mean value). The SIMS data show both systematic variations with time and random Poisson variability, but the former largely cancel out when data for two different isotopes are expressed as a ratio. The precision of a SIMS isotope ratio routinely matches that predicted by Poisson counting statistics and can approach that of conventional bulk analysis techniques for counting times of several hours. All sample analyses must be calibrated for instrumental mass fractionation using SIMS analyses of a standard material. There is often a gradual drift in the mass fractionation with time, but this can be modelled by least-squares regression of the standard isotope ratios. Drift in the sample analyses is eliminated by using the relevant point on this regression line to calibrate each sample. The final precision of a corrected isotope ratio must take into account the scatter in both the sample and the standard data.
Summary It has often been observed that fault‐trace lengths tend to follow a power‐law or Pareto distribution, at least for sufficiently large lengths. A very common method of fitting this type of model to data consists of plotting on log–log axes the number of faults with trace length greater than x against x, and reading off the slope of the resulting approximate straight line. We demonstrate that maximum likelihood is a more efficient and less biased method of estimating the power‐law exponent. A further complication is that this log–log plot is often curved, suggesting that the power‐law distribution is not a complete description of the data. In this paper we review the literature on probability distributions with Pareto behaviour for long trace lengths, but not necessarily for short trace lengths. The Feller–Pareto distribution is an attractive family within this class, with many well‐known statistical distributions as special cases. We use maximum likelihood to fit the Feller–Pareto distribution to a sample of 1034 fault‐trace lengths from the South Yorkshire coalfields. We conclude that the Burr III model superficially provides a satisfactory fit to these data. We also discuss an interpretation of the Feller–Pareto model in terms of a particular type of observational bias on data generated from the power‐law distribution. However, there are a number of complications to be considered. In particular, geometrical sampling biases, stereological effects and spatial structure in the data mean that a rigorous analysis is not straightforward. We suggest ways in which future data collection and analysis may address some of these problems. If our sampling protocols and estimation procedures are adopted, geoscientists should be able to estimate the power‐law exponent more accurately and more objectively than with current ad hoc procedures, and with more direct relevance to strain calculations and other geophysical applications. Furthermore, our recommended method of estimation, maximum likelihood, provides point estimates and associated standard errors of the unknown parameters, and is efficient, consistent and relatively straightforward to apply.
Time series of leafing phenology, air temperature and precipitation along with regional changes in carbon dioxide concentration, are analysed in addressing three questions: `what magnitude of changes in flushing day are likely with climate change?', `which taxa have a chilling requirement?' and `is spring starting earlier?'. Phenological observations of leafing, starting in 1739, are used to calibrate a new model that describes the time of flushing. Statistical methods are used to provide full standard errors on the model coefficients. All 13 tree types studied are found to show an association between flushing date and air temperature. Nine out of the 13 tree types show non-linear temperature behaviour. Model output indicates that climate change has a particularly strong effect on flushing date in oceanic climates, such as that of Britain. The modelled responses can be used to estimate the impact that recent changes in temperature have had on the timing of the start of spring. We find that in any future climate-warming, flushing dates of tree types sensitive to `chilling', for example beech and horse chestnut, are likely to move 30 days, or more, out of synchroneity with those sensitive only to springtime warmth. The new phenological model is also applied to fluctuations in the carbon dioxide mixing-ratio. Changes in the phase of the annual CO2 cycle are isolated and used as evidence for an earlier spring. We make use of a complex demodulation analysis of carbon dioxide, as monitored at 27 stations around the world, to reveal that the timing of the annual CO2 cycle has become steadily earlier in recent decades. Finally we show that the estimates of the effect of rising temperatures on the earliness of spring, made using our new phenological model, are in good agreement with global observations of changes in the timing of the annual carbon dioxide cycle if about 50% of the biomass of the world's terrestrial biosphere is fully temperature sensitive and about 50% sensitive to chilling (or to an equivalent inhibitory effect).
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