\documentclass[twoside]{article}
\usepackage[silent,pdf,hyper]{elxfinal}
\usepackage[T2A]{fontenc}
\usepackage{color}
\usepackage{hyperref}
\journalid{ES}{19}{6}{19}
\citnumber{}{10}
\paperid{2019ES000697}
\papertype{RAR}
\cpright{}{2019}
\lefthead{lushnikov and bogoutdinov}
\righthead{introduction to geophysical distributions}
\received{12}{November}{2019}
\accepted{12}{December}{2019}
\published{17}{December}{2019}
\definecolor{brown}{RGB}{157,23,4}


\authaddr{A. A. Lushnikov, Geophysical Center of the Russian Academy of Sciences (GC RAS), Moscow, Russia. (alex.lushnikov@mail.ru)}


\begin{document}
	
\title{An introduction to geophysical distributions}
	
\author{1}{Lushnikov}{A. A.}{}{1}{}{}
\author{2}{Bogoutdinov}{Sh. R.}{}{1}{2}{}
	
\affil{1}{Geophysical Center of the Russian Academy of Sciences (GC RAS)}{Moscow}{}{Russia}
\affil{2}{Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences (IPE RAS)}{Moscow}{}{Russia}
	
\keywd{Random geophysical processes}
\keywd{distributions}
\keywd{master equation}
\keywd{parametrization of distributions}

	
\abstract{This paper  reviews  the methods of treating the results of geophysical observations  typically met in various geophysical studies.
The main emphasize is given to the sets of data on the phenomena undergone the actions of random factors. These data are normally described by the distributions depending on the nature of the processes.  Among them the magnitude of earthquakes,  the diameters of moon craters, the intensity of solar flares, population of cities, size of aerosol and hydrosol particles, eddies in turbulent water, the strengths of tornadoes and hurricanes and many other things. Irrespective of the causes for their randomness these  manifestations of planetary activity are characterized by  few 	distribution functions like Gauss distribution, lognormal distribution, gamma distribution, and algebraic distributions. Each of these 	distributions contains empirical parameters the values of which depend on the concrete nature of the process. Especially interesting are so 	called ``thick'' distributions with the algebraic tails. Possible  parametrizations of these distributions are discussed.}
	
\section{Introduction}\vspace*{1pc}
	
The random factors playing the central role in various geophysical   processes
insistently demand the application of
stochastic approaches for their study. This is true even for phenomena describing by well based equations, because interactions between
several nonlinear processes often lead to their chaotic behavior. The approaches allowing for consideration such processes had appeared
since  very long ago and have proven their use in all cases without exception [\itc{Ding and Li,} \reflink{Ding07}{2007}; \itc{Eckmann and Ruelle,} \reflink{Eckm85}{1985}; \itc{Van Kampen,} \reflink{Kamp07}{2007}; \itc{Klyatskin,} \reflink{Klya05}{2005}; \itc{McKibben,} \reflink{McKi11}{2011}; \itc{Michael,} \reflink{Mich12}{2012}].\vspace*{.5pc}

In this paper we discuss the geophysical phenomena describing by slowly varying functions of time over significant time intervals rather
than short-period fluctuations, which are in addition small (in amplitude, for example) and from the first sight cannot noticeably affect
the course of the geophysical events. The most bright example of such phenomena is well known in the rock destruction theory, when very small
and almost imperceptible chan\-ges accumulate, and ultimately lead to disruptions in the development of rock material and then  to complete
disruption of the rock [\itc{Eckmann and Ruelle,} \reflink{Eckm85}{1985}; \itc{Michael,} \reflink{Mich12}{2012}].
Among other geophysical phenomena it is necessary to mention one  remarkable mechanism of occurrence of earthquakes. Individual oscillations of
the plates can be small and considered in the harmonic approximation. However, the interactions between the individual modes can lead to their
merger in such a way that the energies of individual modes  are added and a principally new mode forms, which is yet described in the harmonic
approximation, but it  has the energy equal to the sum of the energies of the merged  oscillations. This merger process continues until a single
wave of very large amplitude is  generated. Its energy continues to grow due to the capture of low-energy modes, which  leads to a  catastrophe
(an earthquake, for example).  Similar mechanism can also be responsible for the formation of giant sea waves referred often to as the killer waves.
The dynamics of these accumulation processes depends strongly on the  distributions of the energy carriers [\itc{Bailey,} \reflink{Bail64}{1964}; \itc{Eckmann and Ruelle,} \reflink{Eckm85}{1985}; \itc{Mazo,} \reflink{Mazo02}{2002}; \itc{McKibben,} \reflink{McKi11}{2011}; \itc{Michael,} \reflink{Mich12}{2012}].

Another example is the atmospheric aerosol and its role as the climate formation factor [\itc{Adams and Seinfeld,} \reflink{Adam02}{2002}; \itc{Barrett and Clement,} \reflink{Barr91}{1991}; \itc{Janson et al.,} \reflink{Jans01}{2001}; \itc{Seinfeld and Pandis,} \reflink{Sein98}{1998}].

Generally accepted evolutionary equations more or less correctly describe the {\em average} temporal evolution of complex geophysical structures
(changes in the state of the atmosphere, aerosol processes and climate, the development of territorial structures, demographic structures, etc)
once the transition rates between states of the system are known. For example, the simplest balance equation with empirically introduced birth and
death rates describe the dynamics of transitions between the states with different population sizes. If, however, we introduce the random factor
that spreads the values of the birth and death rates, the dynamic picture can radically change.

\enlargethispage{-.9pc}

\section{Structure of Geophysical Data}

Geophysical data are represented by numerical arrays of dimension $\Omega$. For example, one-dimensio\-nal arrays can represent data on
concentrations of harmful emissions in the atmosphere or a sequence of radar signals from a satellite
at predetermined time intervals, or, for example, randomly selected measurement times.
Our idea is that there is a certain probabilistic process behind geophysical data. Each data sequence is implemented with some probability.
There are several ways to process a signal. This can be, for example, Fourier or wavelet analysis, which allows one to select the main
frequencies of the signal. With this approach, it is assumed that the signal itself is deterministic, and what we observe is a noise +
signal.  Our task is to try to clear the signal from the hindrances. This is precisely the problem that always arises in a location (sound, radio), when
the emitted signal  is known, whereas the incoming signal is composed of noise plus the reflected (partially or completely) signal, by which we should
judge on the geophysical processes occurring far away from the receiver. This is how objects are located by modern positioning systems. The noise introduces
an error for reducing which it is necessary to establish the nature of the noise. In the case of satellite sensing, several such reasons have been
established: we have elucidated the role of Rydberg complexes formed at altitudes of 70 - 110 km in the ionosphere and the role of aerosol particles
exposed to  solar radiation, which leads to their charging. The strong influence of these particles on the state of the ionospheric plasma is
beyond any doubt [\itc{Golubkov et al.,} \reflink{Golu10}{2010}; \reflink{Golu14}{2014}].

Next, there is another type of geophysical signals when the characteristics of the signal generator are not known,
and the task of processing is to extract the maximum information from the signal. For example, it may be a signal from extraterrestrial civilizations.
Then we can expect to find a dominant signal in it and try to filter it out. It is possible to imagine another situation where the signal does not
contain a dominant at all. Attempts to apply some averaging procedures to irregular signal turn out completely unsuccessful [\itc{Bailey,} \reflink{Bail64}{1964}; \itc{Mazo,} \reflink{Mazo02}{2002}].
As an example, the noises from ther Universe  can be mentioned. The question is, then what to do with such signals?

Our probabilistic concept assumes that a stochastic signal is described by the probability of its implementation.
For definiteness, we consider a stationary signal with a random amplitude and introduce a probability to find the given amplitude
in the signal. Our task is to find this probability from the measurement data, which are represented by a set of points belonging to
signals in a given amplitude interval. As the interval narrows and the measurement time lengthens, such a procedure allows one to calculate the
density of the distribution of measurements. However, due to lack of data (and often measurement time), this procedure is low productive.
Therefore, it is more progressive to use cumulative distributions, i.e., to calculate the number of measurements with an amplitude above a given
one. Now, to calculate the distribution density, a discrete mathematical analysis apparatus is required, since we will have to differentiate
a function defined on a discrete set.

As an example, we consider an aerosol counter that determines the concentration of dispersed impurities in atmospheric air [\itc{Julanov et al.,} \reflink{Jula83}{1983}; \itc{Zagaynov et al.,} \reflink{Zaga89}{1989}].
In this counter, each particle enters a countable volume illuminated by a laser beam. The signal scattered from the beam
is read by the photomultiplier. This counter can determine the concentration of particles, provided that the speed of pumping air
through the counting volume is known, and the particles fall into the counting volume one at a time. But a single count cannot be guaranteed,
particles in the counting volume can fall in two, three, etc. This means that the counter will underestimate the counted
concentration of particles. Now the signal (response from the samples) will have a different amplitude depending on the number of
particles falling into the counting volume . Thus, a typical task arises of analyzing a stochastic signal. This task is complicated by
the fact that the aerosol impurity may contain particles of different sizes (polydisperse aerosol).
Then the problem of analysis is complicated, but still possible (see refs [\itc{Adams and Seinfeld,} \reflink{Adam02}{2002}; \itc{Barrett and Clement,} \reflink{Barr91}{1991}; \itc{Julanov et al.,} \reflink{Jula84}{1984}; \reflink{Jula86}{1986}; \itc{Janson et al.,} \reflink{Jans01}{2001}; \itc{Leyvraz,} \reflink{Leyv03}{2003}; \itc{Lushnikov and Kagan,} \reflink{Lush16}{2016}; \itc{Seinfeld and Pandis,} \reflink{Sein98}{1998}; \itc{Schmeltzer et al.,} \reflink{Schm99}{1999}]


\section{Basic Equations}

In what follows we will use the examples adopted from the physics of dispersed particles. We consider the systems comprising the set of particles of different
sizes and will explain the basic idea of the stochastic approach.

\subsection{Particle Size Distributions}

The particle size distributions play the central role in physics and  chemistry of atmospheric aerosol, although a direct observation
of the distributions are possible only in principle. Practically what we really measure is just a response of an
instrument to a given particle size distribution,

\begin{equation}\label{1}
P(x)=\int R(x,a)f(a)da
\end{equation}

\noindent  Here $f(a)$ is the particle size distribution (normally $a$ is the particle radius), $P(x)$ is the reading of the
instrument measuring the property of aerosol $x$, and $R(x,a)$ is referred to as the linear response function of the
instrument. For example, $P(x)$ can be the optical signal from an aerosol particle in the sensing volume of an
optical particle counter, the penetration of the aerosol through the diffusion battery (in this case $x$ is the
length of the battery), or something else. The function $f(a)$ cannot depend on the dimensional variable $a$ alone.
The particle size is measured in some natural units $a_s$. In this case the distribution is a function of $a/a_s$ and
depends on some other dimensionless parameters or groups. The particle size distribution is normalized as follows:

\begin{equation}\label{2}
\int\limits_0^\infty f(a/a_s)\frac{da}{a_s}=1.
\end{equation}

\noindent The length $a_s$ is  a parameter of a distribution.
Although the aerosol particle size distribution is so elusive characteristic of the aerosol, it is still convenient to
introduce it because  all properties of aerosols can be unified in this way.

In some cases the distribution function can be found theoretically on solving  dynamic equations governing the
time
evolution of the particle size distribution, but the methods for analyzing these equations are not yet reliable,
not mentioning the information on the coefficients entering them. This is the reason why the phenomenological
distributions are so widely spread.

\subsection{The Master Equation}

In various geophysical problems it is principally impossible to exclude the influence of random factors. This means that we have to operate with the probability of finding the system in a given state, rather than average time (or ensemble) characteristics of the state. Meanwhile,  ordinary considerations always ignore the deviations from the average picture and operate only with the average characteristics or lower moments.  This fact means that it is necessary to elaborate the efficient methods for consideration of random processes in geophysical systems. For example, global disasters such as earthquakes, tsunamis, tornadoes, epidemics, environmental disasters, etc.  result from a coherent interplay of random factors.
In such cases, the mathematical description should rely upon  other principles than those  widely adopted in modern geophysics. Specifically,
we refuse of the description of the state of the system in terms of time dependent average distributions and apply the description in terms of the occupation numbers.  We exemplify  our idea by considering the set of particles of different size.

\begin{eqnarray*}
Q=n_1,n_2\ldots n_g\ldots
\end{eqnarray*}

\noindent The set changes  through time $t$. For applications however it is enough to have the average spectrum,

\begin{eqnarray*}
\bar n_g(t)=\sum\limits_{n_g}n_g(Q)W(Q,t)\Delta(n_g-n_g(Q))
\end{eqnarray*}

\noindent where $W(n_1,n_2\ldots)$ is the probability of realization of the given set, $n_g(Q)$ denotes the number of the $g$--objects  in the realization $Q$,
$\Delta$ stands for the Kroneker delta symbol. Of course, it is much more convenient to deal with the average spectrum depending on one variable. The problem is then from where to get the probability $W(Q,t)$?

The dynamics of the system depends on the rates of transitions between the states $Q$. Let us assume that these
rates are known and that they depend on the couple of the states $Q^+$ and $Q$. The development of the system is then given by the time sequence

\begin{eqnarray*}
Q^+\to Q\to Q^{-},                             %%%%\label{5}
\end{eqnarray*}

\noindent  i.e., the system jumps from the point of the phase space $Q^+$ to point $Q$, the to $Q^-$ etc.
It is possible to write down the equation (the Master equation) governing the the time evolution of the probability $W(Q,t)$.

\begin{equation}\label{3}
\begin{array}{rl}
\frac{\partial W(Q,t)}{\partial t}=&\sum\limits_{Q^+}A(Q^+,Q)W(Q^+,t)- \\
&W(Q,t)\sum\limits_{Q^-}A(Q,Q^-)
\end{array}                          %%%
\end{equation}

The first term on the right--hand side of this equation is just the rate of transitions from all possible states $Q^+$ to the state $Q$,
the second term is responsible for the losses of the probability because of the jumps to all accessible states $Q^-$.

\section{Examples of Distributions}



We demonstrate first the ideology of the mean field approach, The simplest example is the point moving along $x$ axis with the velocity $v(t)$.
Let the average velocity be $\bar v$ and otherwise the function $v(t)$ is  random. Then the natural desire comes up to write the equation of
motion in the form,

\begin{eqnarray*}
\frac{d\bar x}{dt}=\bar v                   %%%\label{7}
\end{eqnarray*}

\noindent  and to solve it,

\begin{eqnarray*}
\bar x(t)=\bar vt                           %%%\label{8}
\end{eqnarray*}

\noindent  The result can be compared to the exact solution,

\begin{eqnarray*}
x(t)=\int\limits_0^tv(t')dt'                %%%\label{9}
\end{eqnarray*}

\noindent It is clear that the average $\bar x$ can deviate from exact solution very strongly.


Another example operates with the birth-death process. We consider the evolution of the total size $n(t)$ of population in a town. Let us introduce birth and death
rates ($\alpha$ and $\beta$ respectively). Then for the average size
$\bar n(t)$ we can write the equation,

\begin{eqnarray*}
\frac{d\bar n}{dt}=\alpha\bar n-\beta\bar n.            %%\label{10}
\end{eqnarray*}

The solution to this equation is well known,

\begin{eqnarray*}
\bar n(t)=\bar n_0e^{\mu t},
\end{eqnarray*}

\noindent where $\mu=\alpha-\beta$ is the difference between fertility and mortality. If $\mu>0$ the population grows, otherwise ($\mu<0$) it diminishes.
At $\mu=0$ the population does not change and remain equal to $n_0$ -- its initial size.

Now let us introduce a random factor. We assume that $\mu$ is distributed over the Gauss low,

\begin{eqnarray*}
W(\mu)=\sqrt{\frac a\pi}e^{-a\mu^2}             %%%\label{12}
\end{eqnarray*}

\noindent  Here $a$ is a constant. Let us calculate the average size,

\begin{equation}\label{4}
\bar n(t)=n_0\sqrt{\frac a\pi}\int\limits_{-\infty}^\infty e^{-a\mu^2+\mu t}d\mu
\end{equation}

\noindent On integrating over $\mu$  yields,

\begin{eqnarray*}
\bar n(t)=\exp(t^2/2\sqrt a)                %%%\label{13}
\end{eqnarray*}

\noindent  We see that the random factor decisively changes the result. The population grows even if the mortality exceeds the fertility.

Our next example is a particle jumping over 1D lattice. The probability per unit time for a jump between neighboring sites is $1/\tau$. Then according to our ideology we can write,

\begin{eqnarray*}
\frac{dW}{dt}=\frac1\tau(W_{n-1}-W_n)               %%%\label{14}
\end{eqnarray*}

\noindent where $\tau$ has the meaning of average time of a single jump.  The Laplace transform $w_n(p)=\int_0^\infty W_n(t)e^{-pt}$ is determined by the set of algebraic equations,

\begin{equation}\label{5}
pw_n-W_n(0)=\frac1\tau(w_{n-1}-w_n)
\end{equation}

\noindent  where $W_n(0)$ is the initial probability for the particle to occur at the site $n$. We assume that
$W_n(0)=\delta_{n,0}$, i.e., the particle is initially  located at the origin of coordinate.
The set Eq.~\eqref{5} is then readily solved,

\begin{eqnarray*}
w_n(p)=(p+1/\tau)^n                     %%%\label{16}
\end{eqnarray*}

\noindent On inverting this we find,

\begin{eqnarray*}
W_n(t)=\frac{(t/\tau)^n}{n!}e^{-t/\tau}                 %%%\label{17}
\end{eqnarray*}

\noindent This is the Poisson distribution that emerges
instead of $W_n(t)=\Delta(n-n_0(t))$. Here $n_0(t)1/\tau$ is the average velocity of the particle and $\Delta(i-k)$ stands for the Kroneker delta ($\Delta(0)=1$ and $\Delta(x)=0$ at $x\neq0$).


\subsection{Lognormal Distribution}

The lognormal distribution looks as follows,

\begin{equation}\label{6}
f_L(a)=\frac1{\sqrt{2\pi} (a/a_s)\ln\sigma}\exp\left[-\frac1{2\ln^2\sigma}\ln^2\frac a{a_s}\right]
\end{equation}

\noindent  Here $a$ is the particle radius. This distribution depends on  two parameters: $a_s$ and $\sigma$,
where $ a_s$ is the characteristic particle
radius and $\sigma$ ($\sigma>1$) is the width of the distribution. Equation \eqref{6} is known as the lognormal distribution.
It is important to emphasize that it is not derived
from theoretical considerations. Rather, it is introduced by hands. The function $f_L(a)$ is shown in \figref{1}  for different $\sigma$.

\begin{figure}                         % Fig. 1
\figurewidth{20pc}
\setimage{}{}{20pc}{}{2019es000697-f01}
\caption{Shown is the lognormal distributions with  $\sigma=1.5$ (curve 1),  $\sigma=2$ (curve 2), and $\sigma=2.5$ (curve 3). The parameter $\sigma$
defines the width of the distribution. The dimensionless size is defined as $a/a_s$}
\end{figure}

\subsection{Generalized Gamma Distribution}

This size distribution is given by the formula:

\begin{eqnarray*}
f_G(a)=\left(\frac a{a_s}\right)^k\frac j{\Gamma((k+1)/j)}\exp[-(a/a_s)^j]      %%%\label{20}
\end{eqnarray*}

\noindent  Here $\Gamma(x)$ is the Euler gamma--function. The distribution $f_G$ depends on three parameters,
$r_s$, $k$ and $j$. \figref{2} displays
the generalized gamma-distribution for three sets of its parameters.

\begin{figure}                         % Fig. 2
\figurewidth{20pc}
\setimage{}{}{20pc}{}{2019es000697-f02}
\caption{Shown is the gamma--distributions with  three sets of parameters
i. $k=1,\,j=2$, ii. $k=2,\,j=1$, and iii.  $k=5,\,j=2$  (curves 1, 2, and 3 respectively). These parameters
define the shape of the distribution. Again, the dimensionless size is defined as $a/a_s$}
\end{figure}

Once the particle size distributions  are known, it is easy to derive the distribution over the values depending only on the particle size;

\begin{eqnarray*}
f(\psi_0)=\int\delta(\psi_0-\psi(a))f(a)\frac{da}{a_s}              %%%\label{21}
\end{eqnarray*}

\noindent  Here $\delta(x)$ is the Dirac delta function. For example, if we wish
to derive the distribution over the particle masses, then $\psi(a)=(4\pi a^3/3)\rho$ ,
where $\rho$ is the density of the particle material.
Of course, the properties of aerosols depend not only on their size distributions. The shape of aerosol particles and their composition are  important factors.

Lognormal distribution often applies in approximate calculations of important particle formation processes like condensation and coagulation [\itc{Barrett and Clement,} \reflink{Barr91}{1991}; \itc{Friedlander,} \reflink{Frie00}{2000}; \itc{Janson et al.,} \reflink{Jans01}{2001}; \itc{Seinfeld and Pandis,} \reflink{Sein98}{1998}].



\subsection{Scaling and Algebraic Distributions}

\begin{figure}[t]                         % Fig. 3
\figurewidth{20pc}
\setimage{}{}{20pc}{}{2019es000697-f03}
\caption{An example of the distribution with the algebraic tail given by Eq.~\eqref{6}
The dimensionless size is defined as $a/a_s$}
\end{figure}

The distribution $f_\lambda(x)$ possessing the property

\begin{eqnarray*}
f_\lambda(ax)=a^\lambda f_\lambda(x)                %%%\label{22}
\end{eqnarray*}

\noindent  is referred to as the scaling invariant one. The reasons for this are apparent: if we change the scale of $f$
then the distribution preserves its shape but just acquires the multiplier $a^{-\lambda}$. Indeed,
$f(x/a)=a^{-\lambda}f(x)$. The general solution to this equation  is,
\begin{eqnarray*}
f(x)=Ax^{-\lambda}                      %%%\label{23}
\end{eqnarray*}

\noindent  It is seen that this distribution cannot be normalized to 1 at $\lambda\ge1$, because the respective integral diverges. In order to avoid this difficulty a modified distributions is introduced,

\begin{equation}\label{7}
f(x)=A x^{-\lambda}e^{-\gamma x^{-\sigma}}
\end{equation}

\noindent  with $\sigma>0$ and
$$
A^{-1}=\gamma^{(\lambda+\sigma+1)/2}\sigma^{-1}\Gamma(\lambda+1/\sigma)
$$
The  exponential multiplier kills the singularity of $f(x)$ at small $x$. An example of such. distribution is shown in  \figref{3}
Of course, another suitable function can be used instead of the exponent.

The Student distribution is well known to all who deal with the statistical treatment of the result.
This distribution looks as follows,

\begin{eqnarray*}
f(x)=\frac2{\sqrt\pi B(1/2,n/2)}\left(1+\frac{x^2}n\right)^{-(n+1)/2}       %%%\label{24}
\end{eqnarray*}

\noindent  It reveals the algebraic behavior at large $x$. Here $B(x,y)$ is the Euler beta function.  \figref{4} displays the Student distribution for $n=3$.

\begin{figure}[t]                         % Fig. 4
\figurewidth{20pc}
\setimage{}{}{20pc}{}{2019es000697-f04}
\caption{The example of the Student distribution. The parameters are shown in the figure. The dimensionless size is defined as $a/a_s$}
\end{figure}

\subsection{Extended Poisson's Distribution}

Here we consider the simplest process that generates the Poisson distribution. The example given below is adopted from the demography but many similar processes are met in the geophysics.

The Malthus balance (the birth-death process $n-1\to n\to n_1$)  is so primitive that nobody suspects that something unusual can stay behind.
To our great surprise, it is not so, and an alternative consideration
starting with the same idea that the change in population size comes from the difference between the birth and the death rates leads
to rather unusual probability distribution of the sizes, although from the first sight, nothing except for the Poisson distribution is expected.
The reality, however, occurs more complex.

First of all, let us formulate the balance equation for the probability $w(n,t)$ to find $n$ individuals in the population
at time $t$ assuming  the rates
of the birth--death process to be linear in $n$. This equation claims,
\begin{equation}\label{8}
\begin{array}{rcl}
\frac{dw(n,t)}{dt}&=&
\varkappa [(n-1)w(n-1,t)-nw(n,t)]- \\
& &\lambda[ (n+1)w(n+1,t)-nw(n,t)].
\end{array}
\end{equation}


The right--hand side (RHS) of this equation describes the jumps $n-1\longrightarrow n$ (the birth process) and $n+1\longrightarrow n$ (the death process)
that change the population size by $\pm1$ respectively.

In order to solve Eq.~\eqref{8} we introduce the generating function for $w(n,t)$,

\begin{eqnarray*}
F(z,t)=\sum\limits_{n=0}^\infty w(n,t)z^n.              %%%\label{27}
\end{eqnarray*}

\noindent  On multiplying both sides of Eq.~\eqref{4} by  $z^n$ and summing over all $n$ yield the first--order partial differential equation for $F(z,t)$,

\begin{equation}\label{9}
\frac{\partial F}{\partial t}=(z-1)(\varkappa z-\lambda)\frac{\partial F}{\partial z}.
\end{equation}

\noindent  It is easy to check that the general solution to Eq.~\eqref{6} has the form:

\begin{equation}\label{10}
F(z,t)=\psi\left(\frac{1-z}{\lambda-\varkappa z}e^{\mu t} \right).
\end{equation}

\noindent  The function $\psi$ should be determined from the initial condition $F(z,0)=F_0(z)$ or

\begin{eqnarray*}
F_0(z)=\psi\left(\frac{1-z}{\lambda-\varkappa z} \right).               %%%\label{8a}
\end{eqnarray*}

\noindent  We introduce $z(\xi)$ as the solution to the equation $\xi=(1-z)/(\lambda-\varkappa z)$. Hence,

\begin{eqnarray*}
z(\xi)=\frac{1-\lambda\xi}{1-\varkappa\xi}              %%%\label{9a}
\end{eqnarray*}

\noindent  and

\begin{equation}\label{11}
\psi(\xi)=F_0\left(\frac{1-\lambda\xi}{1-\varkappa\xi} \right).
\end{equation}

On substituting $\xi=[(1-z)/(\lambda-\varkappa z)]e^{\mu t}$ from Eq.\eqref{10} into Eq.~\eqref{11} finally yields,

\begin{eqnarray*}
F(z,t)=F_0\left(\frac{\lambda-\varkappa z-\lambda(1-z)e^{\mu t}}{\lambda-\varkappa z-\varkappa(1-z)e^{\mu t}} \right).
\end{eqnarray*}

\noindent  For further analysis it is more convenient to have the generating function $F(z,t)$ in the form:

\begin{equation}\label{12}
F(z,t)=F_0 \left(A+\frac {Qz}{1-\displaystyle\frac z{z_0}} \right).
\end{equation}

\noindent  Here
$$
A=\frac{\lambda(e^{\mu t}-1)}{\varkappa e^{\mu t}-\lambda},\quad Q=\frac{\mu^2 e^{\mu t}}{(\varkappa e^{\mu t}-\lambda)^2},
$$
and
\begin{eqnarray*}
z_0=\frac{\varkappa e^{\mu t}-\lambda}{\varkappa(e^{\mu t}-1)}          %%%\label{13a}
\end{eqnarray*}



It is important to note that above three functions are always positive because $\text{sgn}(e^{\mu t}-1)=\text{sgn}(\varkappa e^{\mu t}-\lambda)
=\text{sgn} (\varkappa-\lambda)$. Here $\text{sgn}(x)=1$ at positive $x$ and $\text{sgn}(x)=-1$ otherwise.


It is easy to find  two first moments of $w(n,t)$. On differentiating Eq.~\eqref{6} once  over $z$ and putting $z=1$ reproduce Eq.~\eqref{11}
for $\bar n(t)=\sum_nnw(n,t)=\partial_zF(1,t)$.
Repeating this operation gives the closed equation for $\partial_{zz}^2F|_{z=1}=\Phi_2(t)=\sum_n(n^2-n)w(n,t)$,

\begin{eqnarray*}
\frac{d\Phi_2}{dt}=2\mu \Phi_2+2\varkappa\bar n.
\end{eqnarray*}

\noindent  The solution to this equation is,
\begin{equation}\label{13}
\Phi_2(t)=\left (\frac{\Phi_{2,0}}{\bar n_0^2}+\frac{2\varkappa}{\mu \bar n_0}\right)\bar n^2(t)-\frac{2\varkappa}\mu\bar n(t).
\end{equation}


Let us analyze  the initial Poisson's distribution.  In this case $\Phi_{2,0}=\bar n_0^2$. We calculate the difference
$\Delta=\overline{n^2}(t)-\bar n^2(t)$. This
difference is readily found from Eq.~\eqref{13},%%% add 12

\begin{eqnarray*}
\Delta(t)=\bar n(t)\left[1+\frac{2\varkappa}\mu(e^{\mu t}-1)\right].     %%%\label{16a}
\end{eqnarray*}

\noindent  It is interesting to note that $\Delta(t)$ grows linearly with time as $\mu\longrightarrow0$, $\Delta(t)=n_0(1+2\varkappa t$), whereas the total
population size does not change.

Of great interest is thus
to find the time dependence of the distribution explicitly. To this end we apply  Eq.~\eqref{12} to the function

\begin{equation}\label{14}             %%%\label{17a}
F_0(z)=e^{\bar n_0(z-1)}.
\end{equation}

\noindent  This initial generating function corresponds to the Poisson distribution Eq.~\eqref{1}


The size distribution $w(n,t)$ is expressed through the contour integral of $F(z,t)$,

\begin{equation}\label{15}                 %%%\label{18a}
w(n,t)=\frac1{2\pi i}\oint\frac{F(z,t)}{z^{n+1}}dz.
\end{equation}

\noindent  The integration goes counterclockwise along the contour surrounding the origin of coordinates in the complex plane $z$.
On applying  Eq.~\eqref{4} to $F_0(z)$ given by Eq.~\eqref{14} and substituting the result into Eq.~\eqref{15} yield,

\begin{equation}\label{16}             %%%\label{19a}
w(n,t)=\frac1{2\pi i}e^{-\bar n_0a(t)}\oint\exp\left(\frac{\bar n_0zQ(t)}{1-\displaystyle\frac z{z_0}} \right)\frac{dz}{z^{n+1}},
\end{equation}

\noindent   where

\begin{eqnarray*}
a(t)=[A(t)-1]=\frac{\mu e^{\mu t}}{\varkappa e^{\mu t}-\lambda}.            %%%\label{20a}
\end{eqnarray*}

\noindent  The function $a(t)\ge0$.

The integration in Eq.~\eqref{16} is readily performed to give

\begin{equation}\label{17}                     %%%%\label{21a}
\frac1{2\pi i}\oint\exp\left(\frac{\bar n_0zQ(t)}{1-\displaystyle\frac z{z_0}} \right)\frac{dz}{z^{n+1}}={\cal L}_n(\bar n_0Qz_0),
\end{equation}

\noindent  where ${\cal L}_n(x)$ are the polynomials of the $n$--th order that can be expressed through Laguerre's  polynomials $L_n(x)$ as follows:

\begin{eqnarray*}
{\cal L}_n(x)=L_n(-z_0\bar n_0Q)-L_{n-1}(-z_0\bar n_0Q),       %%%\label{21b}
\end{eqnarray*}

\noindent  where

\begin{eqnarray*}
L_s(x)=\frac{e^{x}}{s!}\frac{d^s}{dx^s}x^s e^{-x}.                  %%%\label{22a}
\end{eqnarray*}

In deriving Eq.~\eqref{17} we used the fact that

\begin{eqnarray*}
(1-z)^{-1}\exp(xz/(z-1))=\sum\limits_{n=1}^\infty L_n(x)z^n             %%%\label{22b}
\end{eqnarray*}

\noindent  is the generating function for the Laguerre polynomials $L_s(x)$ [\itc{Lushnikov and Kagan,} \reflink{Lush16}{2016}].

From Eqs \eqref{16} and \eqref{17} we finally have,

\begin{eqnarray*}
w(n,t)=e^{-\bar n_0a}z_0^{-n}{\cal L}_n(\bar n_0Qz_0).              %%%\label{23a}
\end{eqnarray*}

\begin{figure}[t]                         % Fig. 5
\figurewidth{17pc}
\setimage{}{}{17pc}{}{2019es000697-f05}
\caption{This figure displays the normal distribution for three sets of parameters. The dimensionless size is defined as $a/a_s$}
\end{figure}

\subsection{Normal Distribution}

Let us consider a collection of sites occupied by $n_1,n_2\ldots n_G$  elements These occupation numbers are considered as random variables. Let the total number of the elements is fixed  and equal to $K$. Let us try to find the probability to find $W(n)$ the probability to have exactly $n$ units in one of the site.

\begin{equation}\label{18}                  %%%%\label{28}
W(n)=\sum\Delta(n_1+n_2+\ldots n_g-K)
\end{equation}

\noindent  where $\Delta(k)=1$ at $k=0$ and $\Delta(k)=0$ otherwise.

The summation on the right-hand side of Eq.~\eqref{18} goes over all sites  except the site with number $k$. Of course, the final result is independent of
$k$. We apply the integral representation of $\Delta(k)$.

\begin{eqnarray*}
\Delta(k)=\frac1{2\pi i}\oint\frac{dz}{z^{k+1}}                 %%%\label{29}
\end{eqnarray*}

\noindent  where the integration contour surrounds the point $z=0$ and the integration goes in the counterclockwise direction. Then we have instead of Eq.~\eqref{18},

\begin{equation}\label{19}                  %%%\label{30}
W(n)=\frac1{2\pi i}\oint\prod\frac{dz}{z^{k+1}}=\frac1{2\pi i} \oint\frac{dz}{z^{K+1}}F^K(z)
\end{equation}

\noindent where

\begin{eqnarray*}
F(z)=\frac1{2\pi i}\oint\frac{dz}{f(z)}                     %%%\label{31}
\end{eqnarray*}

At large $N$ the integral in Eq.~\eqref{19} can be evaluated by the saddle point method.
the result has the well familiar form:

\begin{eqnarray*}
W(n)=\frac1{\sqrt{2\pi\sigma^2  }}\exp\left(-\frac{(x-x_0)^2}{2\sigma}\right) %%\label{32}
\end{eqnarray*}

\noindent Tht normal distribution is shown in \figref{5}.


\subsection{Scaling Distributions}

Let us consider the distributions evolving with time. A simplest example is the diffusion of aerosol particles in the atmosphere.
This process is governed by the well--known diffusion equation,

\begin{equation}\label{20}                  %%%\label{33}
\frac{\partial W}{\partial t}=D\frac{\partial^2W}{\partial x^2}
\end{equation}

\noindent  Here $t$ is time, $x$ is the spatial coordinate of particle, $D$ is the diffusivity of the particle. The solution to this equation should depend a
dimensionless group composed the values entering the diffusion equation. The only dimensionless combination composed of time, coordinate, and the diffusivity is $\zeta=x^2/Dt$. Therefore

\begin{equation}\label{21}              %%%\label{34}
W(x,t)=f(x^2/Dt)
\end{equation}

\noindent  The function $f(\zeta)$ can be found by substituting Eq.~\eqref{21} into Eq.~\eqref{20}. The final answer is,

\begin{equation}\label{22}                  %%%\label{35}
f(\zeta)=\frac1{\sqrt{2\pi\zeta}}e^{\zeta^2/2}.
\end{equation}

A hundred years ago
Smoluchowski (see [\itc{Friedlander,} \reflink{Frie00}{2000}]) formulated his salient equation that describes the coagulation process in aerosols.
Since then the Smoluchowski approach found  wide applications in numerous areas of physics, chemistry, economy,
epidemiology, and many other branches of science [\itc{Adams and Seinfeld,} \reflink{Adam02}{2002}; \itc{Janson et al.,} \reflink{Jans01}{2001}; \itc{Leyvraz,} \reflink{Leyv03}{2003}; \itc{Pruppacher and Klett,} \reflink{Prup06}{2006}; \itc{Schmeltzer et al.,} \reflink{Schm99}{1999}; \itc{Seinfeld and Pandis,} \reflink{Sein98}{1998}; \itc{Seinfeld,} \reflink{Sein08}{2008}].

From the first sight the coagulation process looks rather offenceless, a system of
$M$ monomeric objects begins to evolve by pair coalescence of $g$-- and $l$--mers according to the scheme,

\begin{eqnarray*}
(g)+(l)\longrightarrow(g+l).                        %%%\label{36}
\end{eqnarray*}

And there is not a problem to write down the kinetic equation governing the process, everyone can do it,
\begin{align}\label{23}                %%%%\label{37}
\frac{dc_g}{dt}=I+\frac12\int\limits_0^gK(g-l,l)c_{g-l}c_ldl- \nonumber  \\
c_g\int\limits_{l=0}^\infty K(g,l)c_l dl.
\end{align}

\noindent  This is the famous Smoluchowski's equation.
Here the coagulation kernel $K(g,l)$ is the transition rate for the process given by Eq.~\eqref{22} which is assumed to br a homogeneous
function of its arguments $K(ag,al)=a^\lambda K(g,l)$.
The first term on the right--hand side (RHS) of Eq.~\eqref{2} describes the gain in
the $g$--mer concentration $c_g(t)$ due to coalescence of $(g-l)$-- and $l$--mers, while the second one is responsible for the losses of $g$--mers due to their sticking to all other particles. In what follows we will use the dimensionless version of this equation, i.e., all concentrations are measured
in  units of the initial monomer concentration $c_0$ and the time in  units of $1/c_0K(1,1)$.  More details can be found in
the review article [\itc{Leyvraz,} \reflink{Leyv03}{2003}].

Here we consider a stationary version of this equation: we put $\partial_tc=0$ and, in addition, consider a simplified version of the kernel
$K(g,l)=g^\alpha l^\alpha$. In this case Eq.~\eqref{23} can be solved exactly.
The result is,

\begin{eqnarray*}
c_g=A g^{3/2+\alpha}                %%%\label{38}
\end{eqnarray*}

\noindent  where $A$ is a constant.


\section{Random Factors and Distributions}

The random factors can affect the geophysical processes and change the shapes of the distribution. Here we consider two examples.

\subsection{Birth--Death Process}

Let the population growth is governed by the Malthus equation .

\begin{eqnarray*}
\frac{dn}{dt}=\kappa n-\lambda n=\mu n                  %%%\label{39}
\end{eqnarray*}

\noindent  with   $n(t)$  being the mean population size at time  $t$, $\kappa$ and $\lambda$ are the birth and death rates
respectively and $\mu=\kappa-\lambda$  being the growth rate coefficient. The population size grows at   $\mu>0$  or falls at $\mu<0$)

\begin{eqnarray*}
n=n_0e^{\mu t}                              %%%\label{40}
\end{eqnarray*}

\noindent Let us assume now that the rate coefficient is a random value distributed over the Gauss law

\begin{eqnarray*}
W(\mu)=A\exp[-a(\mu-\mu_0)^2]                       %%%\label{41}
\end{eqnarray*}

\noindent where $A=\sqrt{\pi/a}$  is the normalization coefficient and  $a$  is a dispersion parameter  The distribution $W(\mu)$ is normalized to 1, i.e.,.

\begin{eqnarray*}
\int_{-\infty}^\infty W(\mu)d\mu=1                      %%%\label{42}
\end{eqnarray*}

\noindent We average $n(t)$ over the distribution $W$   %%%, given by eq. (q)
and find,

\begin{eqnarray*}
\bar n(t)=An_0\int_{-\infty}^\infty W(\mu) e^{\mu t}d\mu=n_0 e^{\mu_0 t}e^{t^2/4a}    %%%\label{43}
\end{eqnarray*}

\noindent Hence, irrespective of the value of mean fertility the population swiftly grows
The distribution function for $n$ can also be readily found. It is,

\begin{eqnarray*}
w(n,t)= \sqrt{\frac\pi a}\int_{-\infty}^\infty W(\mu)\delta(n- n_0e^{\mu t})d\mu    %%%\label{44}
\end{eqnarray*}

\noindent The integration gives the lognormal distribution
\begin{eqnarray*}
w(n,t)= \sqrt{\frac\pi a}\frac1{tn}\exp\left[-\frac a{t^2}\ln^2(n/n_0)\right]   %%%%\label{45}
\end{eqnarray*}

\subsection{Logistic Model}

\begin{figure}[t]                         % Fig. 6
\figurewidth{20pc}
\setimage{}{}{20pc}{}{2019es000697-f06}
\caption{Time dependence of the size in the case of logistic distribution. The parameters are shown in the figure.}\vspace*{-.5pc}
\end{figure}

Let us consider now the logistic growth

\begin{eqnarray*}
\frac{dn}{dt}=\kappa n-\beta n^2-\lambda n=\mu n-\beta n^2              %%%\label{46}
\end{eqnarray*}

\noindent It is readily seen that the average population size

\begin{eqnarray*}
\frac n{\mu-\beta n}=Ce^{\mu t}                 %%%\label{47}
\end{eqnarray*}

\noindent or

\begin{eqnarray*}
n(t)=\frac{n_0e^x}{1+\beta n_0tF(x)}                %%%\label{48}
\end{eqnarray*}

\noindent where

\begin{eqnarray*}
F(x)=\frac{e^x-1}x\quad x=\mu t                 %%%\label{50}
\end{eqnarray*}

\noindent Now let us average this over the Gauss distribution centered at
$\mu=\mu_0$.

\begin{eqnarray*}
n(t)=\frac{n_0e^{\mu t}}{1+\beta n_0\frac{1-\exp(\mu t)}\mu}=\frac{n_0e^{\mu t}}{1+\beta n_0tF(\mu t)}               %%%\label{51}
\end{eqnarray*}

\noindent where $F(x)=(1-e^x)/x$. On averaging this gives,

\begin{eqnarray*}
\bar n(t)=n_0\int W(\mu)\frac{e^{\mu t}}{1+\beta n_0tF(\mu t)}dt            %%%\label{52}
\end{eqnarray*}

For the logistic law the result is more complicated than in the case of the Malthus distribution

\begin{eqnarray*}
w(n)=\int W(\mu)\delta\left[n-\frac{n_0e^{\mu t}}{1+\beta n_0tF(\mu t)}\right]d\mu
%%%\label{54}
\end{eqnarray*}

\noindent Time dependence of the population size is shown in \figref{6}.

\section{Conclusion}

We have presented several types of probability distributions widely used in geophysics (and not only) We have classified the geophysical
random processes into two groups.
The first group  operates with random processes having a dominant structure i.e.,those including a deterministic process  possessing all the feature of the
whole process. In this case the task of the statistical study can be reduced to the search of this dominant process and the study
the respective deterministic model underlying it. The randomness in this case reveals itself as corrections to the dominant process and plays the
minor role.
The example of such process is the waves on the oceanic surface or mountains on the Earth surface.

More sophisticated tasks appear in the cases where the dominant absents. The simplest example is the random walks of a billiard ball. Its coordinate of its stops
cannot be predicted,
neither its associated with some regular trajectories. It is clear that when such situations emerge we should attack it
by using the probabilistic approaches

In both these cases the description goes in terms of the distributions. Sometimes these distribution can be found theoretically. If not, then we can chose
a distribution from our collection and to try to match its parameters in such a way that the average characteristics would be reproduced.

\acknowledgement{This work was carried out within the project of the fundamental research program of RAS Presidium No. 2 with financial support by the Ministry of Science and Higher Education of the Russian Federation.}

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