Random Processes with Stationary Increments and Composite Spectra

Abstract

A technique for deriving and interpreting fractal geophysical processes with power-law spectra of a different nature is described. Examples include the energy spectra of atmospheric processes and their role in the mixing of impurities, the frequency spectra of sea wind waves, and the spatial spectra of the surface relief of celestial bodies in the solar system. A.N. Kolmogorov’s works in the early 1930s, which were subsequently developed by his followers A.M. Obukhov, A.S. Monin, A.M. Yaglom, and others, are the most important for this. Kolmogorov’s probabilistic laws serve as a model for the analysis of the processes under consideration by methods of the similarity and dimensionality theory.

APPENDIX

The theory of random (stationary and nonstationary) processes was developed by Kolmogorov [1] and Obukhov, Yaglom, and Monin (see [2]). In a systematic and strict sense, the theory of stationary and related random processes and fields is formulated by Yaglom in [21]. Stationary random processes (respectively, processes with stationary nth increments) are usually considered functions of time. They correspond to statistically homogeneous processes (along with processes with homogeneous nth increments) if they are functions of the spatial argument. For example, if acceleration is a stationary random process, its time integral (velocity) will be a random process with stationary first-order increments (n = 1). Here, if $S\left( \omega \right)$ is the acceleration spectrum, the velocity spectrum is ${{S\left( \omega \right)} \mathord{\left/ {\vphantom {{S\left( \omega \right)} {{{\omega }^{2}}}}} \right. \kern-0em} {{{\omega }^{2}}}}{\text{;}}$ it follows from here that, as $\omega \to 0,$ this spectrum can grow infinitely. In this case, the ordinary spectral Fourier representation of the correlation function has no sense and is therefore replaced by structure function $D_{u}^{{\left( 1 \right)}}\left( \tau \right)$ and its spectral representation is of the Fourier type. Subsequent integration yields the coordinates as a random process with stationary increments of the second order (n = 2), and structure functions ${{D}^{{\left( 2 \right)}}}\left( \tau \right)$ (or, ${{D}^{{\left( 2 \right)}}}\left( r \right){\text{,}}$ as used in this paper) still remain as a working tool. For example, for any n, the velocity structure function,

$$D_{u}^{{\left( n \right)}}\left( \tau \right) = \left\langle {{{{\left| {\Delta _{\tau }^{{\left( n \right)}}u\left( t \right)} \right|}}^{2}}} \right\rangle ,$$

(A1)

where $\Delta _{{\tau }}^{{\left( n \right)}}u\left( t \right)$ is an increment of the nth order:

$$\Delta _{{\tau }}^{n}u\left( t \right) = \sum\limits_{k = 0}^n {{{{\left( { - 1} \right)}}^{n}}C_{n}^{k}u\left( {t - k\tau } \right)} ,$$

(A2)

is a Fourier transform of the corresponding spectrum $S_{u}^{{\left( n \right)}}\left( \omega \right){\text{:}}$

$$D_{u}^{{\left( n \right)}}\left( \tau \right) = {{2}^{n}}\int\limits_{ - \infty }^\infty {{{{\left( {1 - \cos \omega \tau } \right)}}^{n}}S_{u}^{n}\left( \omega \right)d\omega } $$

(A3)

(the angle brackets mean averaging). For a power-law spectrum, there is a simple formula relating it to the corresponding structure function: if ${{D}^{{\left( n \right)}}}\left( \tau \right) = C{{\left| \tau \right|}^{\gamma }},$ we have

$$\begin{gathered} {{S}^{{\left( n \right)}}}\left( \omega \right) = \frac{{{{C}_{1}}}}{{{{{\left| \omega \right|}}^{{\gamma + 1}}}}}, \\ {{C}_{1}} = \frac{C}{{{{2}^{{n + 1}}}\int\limits_0^\infty {{{{\left( {1 - \cos \omega \tau } \right)}}^{n}}{{x}^{{ - n - 1}}}dx} }}. \\ \end{gathered} $$

(A4)

Specifically, for first-order increments (n = 1), we have

$${{C}_{1}} = \frac{C}{{4\int\limits_0^\infty {\left( {1 - \cos x} \right){{x}^{{ - \gamma - 1}}}dx} }} = \frac{{2\pi C}}{{\Gamma \left( {\gamma + 1} \right)\sin \left( {{{\pi \gamma } \mathord{\left/ {\vphantom {{\pi \gamma } 2}} \right. \kern-0em} 2}} \right)}},$$

(A5)

where $0 < \gamma < 2$ [21].

If the first-order increments are not statistically stationary and the second-order increments are stationary (and, accordingly, the derivatives are statistically stationary), the characteristics of these processes are structure function ${{D}^{{\left( 2 \right)}}}\left( \tau \right)$ and the spectrum ${{S}^{{\left( 2 \right)}}}\left( \omega \right)$ (relations (A3) and (A4), where $n = 2,$$2 \leqslant \gamma < 4$). For this case, we derived the formula

$${{C}_{1}} = \frac{C}{{8\int\limits_0^\infty {{{{\left( {1 - \cos x} \right)}}^{2}}{{x}^{{ - {\gamma } - 1}}}dx} }} = \frac{{C\left( {{{2}^{{{\gamma } + 1}}} - {{2}^{3}}} \right)\pi }}{{\Gamma \left( {\gamma + 1} \right)\sin ({{ - \pi \gamma } \mathord{\left/ {\vphantom {{ - \pi \gamma } 2}} \right. \kern-0em} 2})}}$$

(A6)

(the denominator in (A6) is positive for $2 < \gamma < 4$). Table A1 presents the values of coefficient $k = {{{{C}_{1}}} \mathord{\left/ {\vphantom {{{{C}_{1}}} {{{C}_{2}}}}} \right. \kern-0em} {{{C}_{2}}}}$ calculated by formulas (A5) for $0 < \gamma < 2$ and (A6) for $2 \leqslant \gamma < 4.$ For $\gamma = 2,$ passing to the limit, we obtain $k\left( {\gamma = 2} \right) = 5.54.$

Table A1.   Values of $k$

A strict systematic description of the theory of stationary and related random processes and fields can be found in the two-volume monographs by Monin and Yaglom [2] and Yaglom [21].