Many indices for measuring species diversity have been proposed. In this article, a link is noted between a common family of diversity indices and non‐additive statistical mechanics. This makes the Shannon index and the Simpson diversity (or Gini coefficient) special cases of a more general index. The general index includes a parameter q that can be interpreted from a statistical mechanics perspective for systems with an underlying (multi)fractal structure. A q‐generalised version of the Zipf–Mandelbrot distribution sometimes used to characterise rank–abundance relationships may be obtained by maximising this entropy.

Abstract. In this paper, classical surrogate data methods for testing hypotheses concerning nonlinearity in time-series data are extended using a wavelet-based scheme. This gives a method for systematically exploring the properties of a signal relative to some metric or set of metrics. A signal continuum is defined from a linear variant of the original signal (same histogram and approximately the same Fourier spectrum) to the exact replication of the original signal. Surrogate data are generated along this continuum with the wavelet transform fixing in place an increasing proportion of the properties of the original signal. Eventually, chaotic or nonlinear behaviour will be preserved in the surrogates. The technique permits various research questions to be answered and examples covered in the paper include identifying a threshold level at which signals or models for those signals may be considered similar on some metric, analysing the complexity of the Lorenz attractor, characterising the differential sensitivity of metrics to the presence of multifractality for a turbulence time-series, and determining the amplitude of variability of the Hölder exponents in a multifractional Brownian motion that is detectable by a calculation method. Thus, a wide class of analyses of relevance to geophysics can be undertaken within this framework.

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16In the last decade, as computing power has increased, there has been an explosion in the 17 use of eddy-resolving numerical methods in the engineering, earth and environmental

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ABSTRACT. We present an analysis of temporal trends inThe snow-depth and temperature control on these patterns seems significant (R = 0.4-0.6), but is stronger at high frequencies for occurrences, and at lower frequencies for runout altitudes. Occurrences between the northern and southern French Alps are partially coupled (R $ $ 0.4, higher at low frequencies). In the north, the main change-point was an earlier shift in $ $1977, and winter snow depth seems to be the main control parameter. In the south, the main change-point occurred later, $ $1979-84, was more gradual, and trends are more strongly correlated with winter temperature.

A method is presented for generating surrogates that are constrained realizations of a time series but which preserve the local mean and variance of the original signal. The method is based on the popular iterated amplitude adjusted Fourier transform method but makes use of a wavelet transform to constrain behavior in the time domain. Using this method it is possible to test for local changes in the nonlinear properties of the signal. We present an example for a change in Hurst exponent in a time series produced by fractional Brownian motion.