Leonard A. Smith

Uncertainty in predictions of the climate response to rising levels of greenhouse gases.

The range of possibilities for future climate evolution needs to be taken into account when planning climate change mitigation and adaptation strategies. This requires ensembles of multi-decadal simulations to assess both chaotic climate variability and model response uncertainty. Statistical estimates of model response uncertainty, based on observations of recent climate change, admit climate sensitivities—defined as the equilibrium response of global mean temperature to doubling levels of atmospheric carbon dioxide—substantially greater than 5 K. But such strong responses are not used in ranges for future climate change because they have not been seen in general circulation models. Here we present results from the ‘climateprediction.net’ experiment, the first multi-thousand-member grand ensemble of simulations using a general circulation model and thereby explicitly resolving regional details. We find model versions as realistic as other state-of-the-art climate models but with climate sensitivities ranging from less than 2 K to more than 11 K. Models with such extreme sensitivities are critical for the study of the full range of possible responses of the climate system to rising greenhouse gas levels, and for assessing the risks associated with specific targets for stabilizing these levels.

A dynamical model for generating synthetic electrocardiogram signals

A dynamical model based on three coupled ordinary differential equations is introduced which is capable of generating realistic synthetic electrocardiogram (ECG) signals. The operator can specify the mean and standard deviation of the heart rate, the morphology of the PQRST cycle, and the power spectrum of the RR tachogram. In particular, both respiratory sinus arrhythmia at the high frequencies (HFs) and Mayer waves at the low frequencies (LFs) together with the LF/HF ratio are incorporated in the model. Much of the beat-to-beat variation in morphology and timing of the human ECG, including QT dispersion and R-peak amplitude modulation are shown to result. This model may be employed to assess biomedical signal processing techniques which are used to compute clinical statistics from the ECG.

Monte Carlo SSA: Detecting irregular oscillations in the Presence of Colored Noise

Singular systems (or singular spectrum) analysis (SSA) was originally proposed for noise reduction in the analysis of experimental data and is now becoming widely used to identify intermittent or modulated oscillations in geophysical and climatic time series. Progress has been hindered by a lack of effective statistical tests to discriminate between potential oscillations and anything but the simplest form of noise, that is, “white” (independent, identically distributed) noise, in which power is independent of frequency. The authors show how the basic formalism of SSA provides a natural test for modulated oscillations against an arbitrary “colored noise” null hypothesis. This test, Monte Carlo SSA, is illustrated using synthetic data in three situations: (i) where there is prior knowledge of the power-spectral characteristics of the noise, a situation expected in some laboratory and engineering applications, or when the “noise” against which the data is being tested consists of the output of an independently specified model, such as a climate model; (ii) where a simple hypothetical noise model is tested, namely, that the data consists only of white or colored noise; and (iii) where a composite hypothetical noise model is tested, assuming some deterministic components have already been found in the data, such as a trend or annual cycle, and it needs to be established whether the remainder may be attributed to noise. The authors examine two historical temperature records and show that the strength of the evidence provided by SSA for interannual and interdecadal climate oscillations in such data has been considerably overestimated. In contrast, multiple inter- and subannual oscillatory components are identified in an extended Southern Oscillation index at a high significance level. The authors explore a number of variations on the Monte Carlo SSA algorithm and note that it is readily applicable to multivariate series, covering standard empirical orthogonal functions and multichannel SSA.

IMPROVED ANALYTIC CHARACTERIZATION OF ULTRAVIOLET SKYLIGHT

Abstract— We present an improved analytic characterization of diffuse spectral irradiance (skylight) for the wavelength range 280–380 nm and solar zenith angle range from 0 to 85°. The formulas achieve greater accuracy by (a) focusing on ratio representations and (b) adjusting the parameters to the more precise radiative transfer calculations of Dave, Braslau and Halpern.

Intrinsic limits on dimension calculations

The combined influences of boundary effects at large scales and nonzero nearest neighbor separations at small scales are used to compute intrinsic limits on the minimum size of a data set required for calculation of scaling exponents. A lower bound on the number of points required for a reliable estimation of the correlation exponent is given in terms of the dimension of the object and the desired accuracy. A method of estimating the correlation integral computed from a finite sample of a white noise signal is given.

Confidence, uncertainty and decision-support relevance in climate predictions

Over the last 20 years, climate models have been developed to an impressive level of complexity. They are core tools in the study of the interactions of many climatic processes and justifiably provide an additional strand in the argument that anthropogenic climate change is a critical global problem. Over a similar period, there has been growing interest in the interpretation and probabilistic analysis of the output of computer models; particularly, models of natural systems. The results of these areas of research are being sought and utilized in the development of policy, in other academic disciplines, and more generally in societal decision making. Here, our focus is solely on complex climate models as predictive tools on decadal and longer time scales. We argue for a reassessment of the role of such models when used for this purpose and a reconsideration of strategies for model development and experimental design. Building on more generic work, we categorize sources of uncertainty as they relate to this specific problem and discuss experimental strategies available for their quantification. Complex climate models, as predictive tools for many variables and scales, cannot be meaningfully calibrated because they are simulating a never before experienced state of the system; the problem is one of extrapolation. It is therefore inappropriate to apply any of the currently available generic techniques which utilize observations to calibrate or weight models to produce forecast probabilities for the real world. To do so is misleading to the users of climate science in wider society. In this context, we discuss where we derive confidence in climate forecasts and present some concepts to aid discussion and communicate the state-of-the-art. Effective communication of the underlying assumptions and sources of forecast uncertainty is critical in the interaction between climate science, the impacts communities and society in general.

Distinguishing between low-dimensional dynamics and randomness in measured time series

The success of current attempts to distinguish between low-dimensional chaos and random behavior in a time series of observations is considered. First we discuss stationary stochastic processes which produce finite numerical estimates of the correlation dimension and K2 entropy under naive application of correlation integral methods. We then consider several straightforward tests to evaluate whether correlation integral methods reflect the global geometry or the local fractal structure of the trajectory. This determines whether the methods are applicable to a given series; if they are we evaluate the significance of a particular result, for example, by considering the results of the analysis of stochastic signals with statistical properties similar to those of observed series. From the examples considered, it is clear that the correlation integral should not be used in isolation, but as one of a collection of tools to distinguish chaos from stochasticity.

Evaluating Probabilistic Forecasts Using Information Theory

Abstract The problem of assessing the quality of an operational forecasting system that produces probabilistic forecasts is addressed using information theory. A measure of the quality of the forecasting scheme, based on the amount of a data compression it allows, is outlined. This measure, called ignorance, is a logarithmic scoring rule that is a modified version of relative entropy and can be calculated for real forecasts and realizations. It is equivalent to the expected returns that would be obtained by placing bets proportional to the forecast probabilities. Like the cost–loss score, ignorance is not equivalent to the Brier score, but, unlike cost–loss scores, ignorance easily generalizes beyond binary decision scenarios. The use of the skill score is illustrated by evaluating the ECMWF ensemble forecasts for temperature at Londons Heathrow airport.

Combining dynamical and statistical ensembles

Aprediction accompanied by quantitative estimates of the likely forecast accuracy is inherently superiorto a single “best guess” forecast. Such estimates can be obtained by “dressing” a single forecast usinghistorical error statistics. Dressing ensemble forecasts is more complicated, as one wishes to avoiddouble counting forecast errors due, for example, to uncertainty in the initial condition when thatuncertainty is explicitly accounted for by the ensemble (which has been generated with multiple initialconditions). The economic value of dressed forecasts has been demonstrated by previous studies. Thispaper presents a method for dressing ensembles of any size, thus enabling valid comparisons to bemade between them. The method involves identifying the “best member” of an ensemble in a multidimensionalforecast space. The statistics of the errors of these best members are used to dress individualforecasts in an ensemble. The method is demonstrated using ECMWF ensemble forecasts, which arecompared with the ECMWF high-resolution best guess forecasts. It is shown that the dressed ECMWFensembles have skill relative to the dressed ECMWF best guess, even at the maximum lead time ofthe ECMWF forecasts (10 days). The approach should be applicable to general ensemble forecasts(initial condition, multi-model, stochastic model etc.), allowing better informed decisions on forecastaquisition and forecast system development.

Identification and prediction of low dimensional dynamics

This contribution focuses upon extracting information from dynamic reconstructions of experimental time series data. In addition to the problem of distinguishing between deterministic dynamics and stochastic dynamics, applied questions, such as the detection of parametric drift, are addressed. Nonlinear prediction and dimension algorithms are applied to geophysical laboratory data, and the significance of these results is established by comparison with results from similar surrogate series, generated so as not to contain the property of interest. A global nonlinear predictor is introduced which attempts to correct systematic bias due to the inhomogeneous distribution of data common in strange attractors. Variations in the quality of predictions with location in phase space are examined in order to estimate the uncertainty in a forecast at the time it is made. Finally, the application of these methods to truly stochastic systems is discussed and the distinction between deterministic, stochastic, and low dimensional dynamics is considered.