From Fractals to Stochastics: Seeking Theoretical Consistency in Analysis of Geophysical Data

Abstract

Fractal-based techniques have opened new avenues in the analysis of geophysical data. On the other hand, there is often a lack of appreciation of both the statistical uncertainty in the results and the theoretical properties of the stochastic concepts associated with these techniques. Several examples are presented which illustrate suspect results of fractal techniques. It is proposed that concepts used in fractal analyses are stochastic concepts and the fractal techniques can readily be incorporated into the theory of stochastic processes. This would be beneficial in studying biases and uncertainties of results in a theoretically consistent framework, and in avoiding unfounded conclusions. In this respect, a general methodology for theoretically justified stochastic processes, which evolve in continuous time and stem from maximum entropy production considerations, is proposed. Some important modelling issues are discussed with focus on model identification and fitting often made using inappropriate methods. The theoretical framework is applied to several processes, including turbulent velocities measured every several microseconds, and wind and temperature measurements. The applications show that several peculiar behaviours observed in these processes are easily explained and reproduced by stochastic techniques.

Appendices

Proof of Infeasibility of Too Steep Slopes in Power Spectrum at Low Frequencies

This proof is summarized here from Koutsoyiannis (2013b) and Koutsoyiannis et al. (2013).

Let us assume the contrary, i.e., that for frequency range 0 ≤ w ≤ ε (with ε however small) the log-log derivative is s ^#(w) = β, or else s(w) = α w ^β where α and β are constants, with β <  − 1. As a result of (2) and (4) the climacogram is related to power spectrum by:

$$ \gamma (k)=\underset{0}{\overset{\infty }{\int }}s(w)\ {\mathrm{sinc}}^2\left(\uppi wk\right)\mathrm{d}w \vspace*{-3pt} $$

(30)

The sinc² function within the integral takes significant values only for w < 1/k (cf. Papoulis 1991, p. 433). Assuming a scale k ≫ 1/ε,

$$ \gamma (k)=\underset{0}{\overset{\infty }{\int }}s(w){\mathrm{sinc}}^2\left(\uppi wk\right)\mathrm{d}w\approx \underset{0}{\overset{\varepsilon }{\int }}{\alpha w}^{\beta }\ {\mathrm{sinc}}^2\left(\uppi wk\right)\mathrm{d}w \vspace*{-3pt} $$

(31)

On the other hand, it can be easily seen that, for 0 < w < 1/k, the following inequality holds:

$$ \operatorname{sinc}\left(\uppi wk\right)\ge 1- wk\ge 0 $$

(32)

Since ε ≫ 1/k, while the function in the integral (31) is nonnegative,

$$ \gamma (k)\approx \underset{0}{\overset{\varepsilon }{\int }}{\alpha w}^{\beta }{\mathrm{sinc}}^2\left(\uppi wk\right)\mathrm{d}w\ge \underset{0}{\overset{1/k}{\int }}{\alpha w}^{\beta }{\mathrm{sinc}}^2\left(\uppi wk\right)\mathrm{d}w\ge \underset{0}{\overset{1/k}{\int }}{\alpha w}^{\beta }{\left(1- wk\right)}^2\mathrm{d}w $$

(33)

By substituting ω = wk into Eq. (33), we find:

$$ \gamma (k)\ge a{k}^{-\beta -1}\underset{0}{\overset{1}{\int }}{\omega}^{\beta }{\left(1-\omega \right)}^2\mathrm{d}\omega $$

(34)

To evaluate the integral in (34) we take the limit for q → ∞ of the integral:

$$ B(q):= \underset{1/q}{\overset{1}{\int }}{\omega}^{\beta }{\left(1-\omega \right)}^2\mathrm{d}\upsilon =\frac{1-q^{-1-\beta }}{1+\beta }-2\frac{1-{q}^{-2-\beta }}{2+\beta }+\frac{1-q^{-3-\beta }}{3+\beta } $$

(35)

Clearly, the limit of B(q) as q → ∞ depends on that of the term with the highest exponent, i.e., q ^{−1 – β}. For β < −1 this term diverges and thus, B(0) = +∞. Then, by virtue of the inequality (34), γ(k)= ∞. For a (mean) ergodic processes γ(k) should necessary tend to 0 for k → ∞ (Papoulis 1991, p. 429). Therefore, the process is nonergodic.

It is interesting to note here that, when |β| < 1, the integral in (31) can be evaluated to give:

$$ \gamma (k)\approx \alpha \underset{0}{\overset{\infty }{\int }}{w}^{\beta }\ {\mathrm{sinc}}^2\left(\uppi w\varDelta \right)\mathrm{d}w=\frac{\alpha \Gamma \left(1+\beta \right)\operatorname{sinc}\left(\uppi \beta /2\right)}{2\left(1-\beta \right){\left(2\uppi \right)}^{\beta }{k}^{1+\beta }} $$

(36)

Clearly, for k → ∞, the last expression gives γ(k) → 0 and thus for |β| < 1 the process is mean ergodic.

This analysis for β < −1 generalizes a result by Papoulis (1991, p. 434) who shows that an impulse at w = 0 corresponds to a non-ergodic process.

Literature Review on the Distribution Function of Wind Speed

A large variety of distributions in the literature (with the most common to be Gaussian, gamma, Weibull, lognormal, Pareto and generalizations thereof as well as mixtures with each other) show equally good agreement with atmospheric wind measurements recorded at different sites around the globe with different climatic conditions.

A sample of recent publications is listed in Table 2 along with the proposed distributions. However, some distributions seem to exhibit good agreement with data at the left or right tail mostly due to different lengths of the examined time series, while arguably most distributions do not exhibit good agreement for the whole range.

Table 2 Recent publications on the distribution function of wind speed