###### Abstract

Experimental data from the National Air Surveillance Network of Japan from
1974 to 1996 and from independent measurements performed simultaneously in the
regions of Ljubljana (Slovenia), Odessa (Ukraine) and the Ukrainian
“Academician Vernadsky” Antarctic station (64∘15W; 65∘15S), where the air
elemental composition was determined by the standard method of atmospheric
particulate matter (PM) collection on nucleopore filters and subsequent
neutron activation analysis, were analyzed. Comparative analysis of different
pairs of atmospheric PM element concentration data sets, measured in different
regions of the Earth, revealed a stable linear (on a logarithmic scale)
correlation, showing a power law increase of every atmospheric PM element mass
and simultaneously the cause of this increase – fractal nature of atmospheric
PM genesis. Within the framework of multifractal geometry we show that the
mass (volume) distribution of atmospheric PM elemental components is a log
normal distribution, which on a logarithmic scale with respect to the random
variable (elemental component mass) is identical to normal distribution. This
means that the parameters of two-dimensional normal distribution with respect
to corresponding atmospheric PM-multifractal elemental components measured in
different regions, are a priory connected by equations of direct and inverse
linear regression, and the experimental manifestation of this fact is the
linear correlation between the concentrations of the same elemental components
in different sets of experimental atmospheric PM data.

###### keywords:

Atmospheric aerosols; Multifractal; Neutron activation analysis; South Pole;
Ukrainian Antarctic station

10.3390/—— xx Received: xx / Accepted: xx / Published: xx On the Fractal
Mechanism of Interrelation Between the Genesis, Size and Composition of
Atmospheric Particulate Matters in Different Regions of the Earth Vitaliy D.
Rusov 1,⋆, Radomir Ilic 2, Radojko R. Jacimovic 2, Vladimir N. Pavlovich 3,
Yuriy A. Bondarchuk 1, Vladimir N. Vaschenko 4, Tatiana N. Zelentsova 1,
Margarita E. Beglaryan 1, Elena P. Linnik 1, Vladimir P. Smolyar1, Sergey I.
Kosenko 1 and Alla A. Gudyma 4 E-Mail<EMAIL_ADDRESS>Tel.: +38-048-2641672;
Fax: ++38-048-2641672.

##  1 Introduction

Analysis of the concentrations of elements characteristics of the terrestrial
crust, anthropogenic emissions and marine elements used in monitoring of the
levels of atmospheric aerosol contamination indicates that these levels at the
two extremes of: (i) the Antarctic (South Pole ref-1 ), and (ii) the global
constituent of atmospheric contamination measured at continental background
stations ref-2  display similar patterns. A distinction between them is,
however, evident and consists in that the mean element concentrations in the
atmosphere over continental background stations ($C_{CB}$) located in
different regions of the Earth exceed the corresponding concentrations at the
South Pole ($C_{SP}$) by some $20-10^{3}$ times.

Comparing the concentrations of a given element i in atmospheric aerosol from
samples $\left\\{C_{SP,i}\right\\}$ and $\left\\{C_{CB,i}\right\\}$, it became
evident that the dependence of the mean concentrations of any particular
element from the sample $\left\\{C_{SP,i}\right\\}$ or
$\left\\{C_{CB,i}\right\\}$ on a logarithmic scale is described by a linear
one, with good precision for any of the elements from a given pair of sampling
station:

$\ln{C_{CB}^{i}}=a_{CB-SP}+b_{CB-SP}\ln{C_{SP}^{i}},$ (1)

$b_{CB-SP}\approx 1,$ (2)

where $a_{CB-SP}$ and $b_{CB-SP}$ are the intercept and slope (regression
coefficient) of the regression line, respectively.

This unique and rather unexpected result was first established by Pushkin and
Mikhailov ref-2 . It is noteworthy, according to ref-2 , that the reason for
the large enrichment of atmospheric aerosol with those elements which are
exceptions to the linear dependence, is related mostly either to the
anthropogenic contribution produced by extensive technological activity, e.g.
the toxic elements (Sb, Pb, Zn, Cd, As, Hg), or to the nearby sea or ocean or
sea as a powerful source of marine aerosol components (Na, I, Br, Se, S, Hg).

However, our numerous experimental data specify persistently that the Pushkin-
Mikhailov dependence (1) in actual fact is the particular case (at $b_{12}=1$)
of more general of linear regression equation

$\ln\left(C_{1i}/\rho_{i}\right)=a_{12}+b_{12}\ln{\left(C_{2i}/\rho_{i}\right)}$
(3)

where $C_{i}$ and $\rho_{i}$ are the concentration and specific density of
i-th isotope component in atmospheric PM measured in different regions (the
indexes 1 and 2) of the Earth.

It is obvious, if we will be able to prove that the linear relation (3) in
element concentrations between the above mentioned samples reflects the more
general fundamental dependence, it can be used for the theoretical and
experimental comparison of atmospheric PM independently of the given region of
the Earth. Moreover, the linear relation (3) can become a good indicator of
the elements defining the level of atmospheric anthropogenic pollution, and
thereby to become the basis of method for determining a pure air standard or,
to put it otherwise, the standard of the natural level of atmospheric
pollution of different suburban zones. This is also indicated by the power law
character of Eq. (3), reflecting the fact that the total genesis of non-
anthropological (i.e natural) atmospheric aerosols does not depend upon the
geography of their origin and is of a fractal nature.

In our opinion, this does not contradict the existing concepts of microphysics
of aerosol creation and evolution ref-3 , if we consider the fractal structure
of secondary ($D_{p}>1\mu m$) aerosols as structures formed on the prime
inoculating centres, ($D_{p}<1\mu m$). Such a division of aerosols into two
classes – primary and secondary ref-3  – is very important since it plays the
key role for understanding of the fractal mechanism of secondary aerosol
formation, which show scaling structure with well-defined typical scales
during aggregation on inoculating centres (primary aerosols) ref-4 .

The objective of this work was twofold: (i) to prove reliably the linear
validity of Eq. (3) through independent measurements with good statistics
performed at different latitudes and (ii) to substantiate theoretically and to
expose the fractal mechanism of interrelation between the genesis, size and
composition of atmospheric PM measured in different regions of the Earth, in
particular in the vicinity of Odessa (Ukraine), Ljubljana (Slovenia) and the
Ukrainian Antarctic station “Academician Vernadsky” (64∘15W; 65∘15S).

##  2 The linear regression equation and experimental data of National Air
Surveillance Network of Japan

In this study, experimental data ref-5  from the National Air Surveillance
Network (NASN) of Japan for selected crustal elements (Al, Ca, Fe, Mn, Sc and
Ti), anthropogenic elements (As, Cu, Cr, Ni, Pb, V and Zn) and a marine
element (Na) in atmospheric particulate matter obtained in Japan for 23 years
from 1974 to 1996 were evaluated. NASN operated 16 sampling stations (Nopporo,
Sapporo, Nonotake, Sendai, Niigata, Tokyo, Kawasaki, Nagoya, Kyoto-Hachiman,
Osaka, Amagasaki, Kurashiki, Matsue, Ube, Chikugo-Ogori and Ohmuta) in Japan,
at which atmospheric PM were regularly collected every month by a low volume
air sampler and analyzed by neutron activation analysis (NAA) and X-ray
fluorescence (XRF). During the evaluation, the annual average concentration of
each element based on 12 monthly averaged data between April (beginning of
financial year in Japan) and March was taken from NASN data reports. The long-
term (23 years) average concentrations were determined from the annual average
concentration of each element.

Analysis of the NASN data ref-5  shows that the highest average concentrations
were observed in Kawasaki (Fe, Ti, Mn, Cu, Ni and V), Osaka (Na, Cr, Pb and
Zn), Ohmuta (Ca) and Niigata (As), respectively. These cities are either
industrial or large cities of Japan. Conversely, the lowest average
concentrations were noticed in Nonotake (Al, Ca, Fe, Ti, Cu, Cr, Ni, V and Zn)
and Nopporo (Mn, As and Pb), as expected ref-5 . On the basis of these
results, Nonotake and Nopporo were selected as the baseline-remote area in
Japan.

A simple model of linear regression, in which the evaluations were made by the
least squares method ref-6 ; ref-7 , was used to build the linear dependence
described by Eq. (3). The results of NASN data presented on a logarithmic
scale relative to the data of Nonotake $\left\\{C_{Nonotake}^{i}\right\\}$
(see Figure 1) show with high confidence the adequacy of experimental and
theoretical dependence of Eq. (3) type. As can be seen from Figure 1, the
Nonotake station was chosen as the baseline, where the lowest concentrations
of crustal and anthropogenic elements and small variations in time (23 years)
were observed ref-5 . The NASN data are presented in the form of “city – city”
concentration dependences.

Figure 1: Relation between annual concentrations of elements in atmospheric
particulate matter over Japan and the same data obtained in the region of
Nonotake (data from NASN of Japan ref-5 ). In some cases (see text) the
concentration of anthropogenic element _Cr_ was excluded from the data. The
underlined cities are large industrial centres in the Japan ref-5 .

Analysis of concentration data for Japanese city atmospheric PM unambiguously
shows that the i-th element mass in atmospheric PM grows by power low, proving
the assumption ref-4 ; ref-8 ; ref-9 ; ref-10  about the fractal nature of
atmospheric PM genesis.

##  3 The linear regression equation and the composition of atmospheric
aerosols in different regions of the Earth

It is evident that in order to generalize the results from NASN data
processing more widely, the validity of Eq. (3) should be checked on the basis
of atmospheric aerosol studies in performed independent experiments at
different latitudes. For this reason such studies were performed in the
regions of Odessa (Ukraine), Ljubljana (Slovenia) and the Ukrainian
Academecian Vernadsky Antarctic station (64∘15W; 65∘15S). The determination of
the element composition of the atmospheric air in these experiments was
performed by the traditional method based on collection of atmospheric aerosol
particles on nucleopore filters with subsequent use of $k_{0}$-instrumental
neutron activation analysis. Regression analysis was used for processing of
the experimental data.

###  3.1 Experimental

####  3.1.1. Collection of atmospheric aerosol particles on nucleopore filters

For collection of atmospheric aerosol particles on filters, a device of the
PM10 type was used with the Gent Stacked Filter Unit (SFU) interface ref-11 ;
ref-12 . The main part of this device is a flow-chamber containing an
impactor, the throughput capacity of which is equivalent to the action of a
filter with an aerodynamic pore diameter of 10 $\mu m$ and a 50% aerosol
particle collection efficiency based on mass, and an SFU interface designed by
the Norwegian Institute for Air Research (NILU) and containing two filters
(Nucleopore) each 47 mm in diameter, the first filter with a pore diameter of
8 $\mu m$ and the second filter with 0.4 $\mu m$ pore diameter. It was
experimentally found ref-12  that such geometry (Figure 2) results in an
aerosol collection efficiency of the first filter of approximately 50%,
whereas for the second filter this value was close to 100% ref-12 . More
detailed results of thorough testing of a similar device can be found in
ref-12 .

Figure 2: Schematic representation of the flow-chamber of the PM10 type device
for air sampling.

####  3.1.2. $k_{0}$-instrumental neutron activation analysis

Airborne particulate matter (APM) loaded filters were pelleted with a manual
press to a pellet of 5 mm diameter and each packed in a polyethylene ampoule,
together with an Al-0.1%Au IRMM-530 disk 6 mm in diameter and 0.2 mm thickness
and irradiated for short irradiations (2-5 min) in the pneumatic tube (PT) of
the 250 kW TRIGA Mark II reactor of the J. Stefan Institute at a thermal
neutron flux of $3.5\cdot 10^{12}cm^{-2}s^{-1}$, and for longer irradiations
in the carousel facility (CF) at a thermal neutron flux of $1.1\cdot
10^{12}cm^{-2}s^{-1}$ (irradiation time for each sample about 18-20 h). After
irradiation, the sample and standard were transferred to clean 5 mL
polyethylene mini scintillation vial for gamma ray measurement. To determine
the ratio of the thermal to epithermal neutron flux ($f$) and the parameter
$\alpha$, which characterizes the degree of deviation of the epithermal
neutron flux from the $1/E$-law, the cadmium ratio method for multi monitor
was used ref-13 . It was found that $f=32.9$ and $\alpha=-0.026$ in the case
of the PT channel, and $f=28.7$ and $\alpha=-0.015$ for the CF channel. These
values were used in calculation of the concentrations of short- and long-lived
nuclides.

$\gamma$-activity of irradiated samples were measured on two HPGe-detectors
(ORTEC, USA) of 20 and 40% measurement efficiency ref-13 . Experimental data
obtained on these detectors were fed into and processed on EG&G ORTEC Spectrum
Master and Canberra S100 high-velocity multichannel analyzers, respectively.
To calculate net peak areas, HYPERMET-PC V5.0 software was used ref-14 ,
whereas for evaluation of elemental concentrations in atmospheric aerosol
particles, KAYZERO/SOLCOI® software was used ref-15 . More details of the
$k_{0}$-instrumental neutron-activation analysis applied could be found in
ref-13 .

###  3.2 Comparative analysis of atmospheric PM composition in different
regions of the Earth

The results of presenting the atmospheric PM concentration values of Ljubljana
$\left\\{C_{Ljubljana}^{i}\right\\}$ and Odessa
$\left\\{C_{Odessa}^{i}\right\\}$ on a logarithmic scale relative to the
similar data from Academician Vernadsky station
$\left\\{C_{Ant.station}^{i}\right\\}$ demonstrate with high reliability that
the correlation coefficient r is approximately equal to unity both for the
direct and reverse regression lines “Odessa - Antarctic station”, “Ljubljana -
Antarctic station” (Figure 3):

$r=\left[b_{12}b_{21}\right]^{1/2}\approx 1,$ (4)

where $b_{12}$ and $b_{21}$ are the slopes of direct and reverse regression
lines (Eq. (1)) for corresponding pairs.

Figure 3: Lines of direct regression “Odessa - Antarctic station” (a),
“Ljubljana - Antarctic Station” (c) and line of inverse regression “Antarctic
Station - Odessa” (b), “Antarctic Station - Ljubljana” (d).

Figures 4 and 5 show the regression lines for daily normalized average
concentrations of crustal, anthropogenic and marine elements (Table
LABEL:tab1), measured on March 2002 in the regions of Odessa (Ukraine), the
Ukrainian Antarctic station (∘15W; 65∘15S) and Ljubljana (Slovenia) relative
to the same data obtained in Nonotake ref-5  (Figure 4) and the South Pole
ref-1  (Figure 5).

Figure 4: Relationship between atmospheric PM elemental concentrations measured in the regions Odessa (Ukraine), Ljubljana (Slovenia), Vernadsky station (64∘15W; 65∘15S), South Pole ref-1  and the same data measured in the region of Nonotake ref-5 . Figure 5: Relationship between atmospheric PM elemental concentrations measured in the regions Odessa (Ukraine), Ljubljana (Slovenia), Vernadsky station (64∘15W; 65∘15S), Nonotake ref-5  and the same data measured in the region of the South Pole ref-1 . Table 1:  Chemical element composition in atmospheric PM. Symbols $C_{SP}$, $C_{CB}$, $C_{Nonotake}$ denote concentrations at the South Pole, continental background stations and Nonotake-city, respectively. Measured concentrations in the vicinity of the Ukrainian Antarctic Station, Odessa and Ljubljana are denoted by $C_{Antarctica}$, $C_{Odessa}$ and $C_{Ljubljana}$ Element | $C_{SP}$ | $C_{CB}$ | $C_{Nonotake}$ | $C_{Antarctica}$ | $C_{Odessa}$ | $C_{Ljubljana}$
---|---|---|---|---|---|---
($ng/m^{3}$)
S | 49 | - | - | - | - | -
Si | - | - | - | - | - | -
Cl | 2.4 | 90 | - | - | - | 528.2
Al | 0.82 | 1.2 $\times$ 103 | 237.4 | - | - | 1152
Ca | 0.49 | - | 185.8 | 93 | 2230 | 1630
Fe | 0.62 | 1 $\times$ 10 2 | 157.6 | 53.4 | 1201 | 1532
Mg | 0.72 | - | - | - | - | 1264
K | 0.68 | - | - | 24.4 | 502.8 | 918
Na | 3.3 | 1.4 $\times$ 10 2 | 538.8 | 361.95 | 393.8 | 1046
Pb | - | 10 | 23.2 | - | - | -
Zn | 3.3 $\times$ 10-2 | 10 | 38.3 | 4.48 | 62.6 | 127.2
Ti | 0.1 | - | 15.4 | - | - | -
F | - | - | - | - | - | -
Br | 2.6 | 4 | - | 1.34 | 14.81 | 9.62
Cu | 2 $\times$ 10 -2 | 3 | 8.20 | 11050 | 4240 | -
Mn | 1.2 $\times$ 10 -2 | 3 | 6.61 | - | - | 34.7
Ni | - | 1 | 1.48 | - | - | -
Ba | 1.6 $\times$ 10 -2 | - | - | - | - | 6.91
V | 1.3 $\times$ 10 -3 | 1 | 2.44 | - | - | 17.08
I | 0.74 | - | - | - | - | 36.24
Cr | 4 $\times$ 10 -2 | 0.8 | 1.14 | 3.11 | 45.2 | -
Sr | 5.2 $\times$ 10 -2 | - | - | - | - | -
As | 3.1 $\times$ 10 -2 | 1 | 2.74 | - | 1.83 | 4.18
Rb | 2 $\times$ 10 -3 | - | - | - | - | 0.92
Sb | 8 $\times$ 10 -4 | 0.5 | - | 0.07 | 1.97 | 4.62
Cd | < 1.5 $\times$ 10 -2 | 0.4 | - | - | - | -
Mo | 49 | - | - | 1.46 | 4.08 | -
Se | < 0.8 | 0.3 | - | 0.02 | 0.47 | 1.21
Ce | 4 $\times$ 10 -3 | - | - | - | 4.22 | 2.72
Hg | 0.17 | 0.3 | - | 0.36 | 2.75 | -
W | 1.5 $\times$ 10 -3 | - | - | 0.24 | - | -
La | 4.5 $\times$ 10 -4 | - | - | - | 1.79 | 1.36
Ga | < 1 $\times$ 10 -3 | - | - | - | - | -
Co | 5 $\times$ 10 -4 | 0.1 | - | 0.02 | 0.89 | 0.29
Ag | < 4 $\times$ 10 -4 | - | - | 2.74 | 1.42 | -
Cs | 1 $\times$ 10 -4 | - | - | - | 0.09 | -
Sc | 1.6 $\times$ 10 -4 | 5 $\times$ 10 -2 | 0.04 | 3.36 $\times$ 10 -3 | 0.21 | 0.298
Th | 1.4 $\times$ 10 -4 | - | - | - | 0.29 | 0.060
U | - | - | - | - | 0.07 | -
Sm | 9 $\times$ 10 -5 | - | - | 2.68 $\times$ 10 -3 | 0.24 | 0.198
In | 5 $\times$ 10 -5 | - | - | - | - | -
Ta | 7 $\times$ 10 -5 | - | - | - | - | -
Hf | 6 $\times$ 10 -5 | - | - | - | - | -
Yb | < 0.05 | - | - | - | - | -
Eu | 2 $\times$ 10 -5 | - | - | - | - | -
Au | 4 $\times$ 10 -5 | - | - | 1.07 $\times$ 10 -5 | 0.006 | -
Lu | 6.7 $\times$ 10 -6 | - | - | - | - | -

The choice of the atmospheric aerosol concentration values of the South Pole
as a baseline, relative to which a dependence of the type given by Eq. (1) was
analyzed, was made for three reasons: (i) these data were obtained by the same
technique and method as in section 3.1, and simultaneously expand the
geographical comparison, (ii) the element spectrum that characterizes the
atmosphere of South Pole covers a wide range of elements (see Table 1); and
(iii) the South Pole has the purest atmosphere on Earth, making it a
convenient basis for comparative analysis.

We present also the monthly normalized average concentrations of atmospheric
PM measured over the period 2006-2007 in the region of the Ukrainian Antarctic
station “Academician Vernadsky” (Figures 6, 7).

Figure 6: The regression lines for the normalized monthly average
concentrations of atmospheric PM measured in the region of the Ukrainian
Antarctic station “Academician Vernadsky” measured in August–December 2006
with respect to October 2006. Figure 7: The regression lines for the
normalized monthly average concentrations of atmospheric PM measured in the
region of the Ukrainian Antarctic station “Academician Vernadsky” in January-
March 2007 with respect to October 2006.

Comparative analysis of experimental sets of normalized concentrations of
atmospheric aerosol elements measured in our experiments and independent
experiments of the Japanese National Air Surveillance Network (NASN) shows a
stable linear (on a logarithmic scale) dependence on different time scales
(from average daily to annual). That points to a power law increase of every
atmospheric PM element mass (volume) and simultaneously to the cause of this
increase – the fractal nature of atmospheric PM genesis.

In other words, stable fulfillment of the equality (4) not only for the
experimental data shown in Figures 3-7, but also for any pairs of the NASN
data ref-5  (see Figure 2) unambiguously indicates that, on the one hand, the
model of linear regression is satisfactory and, on the other hand, any of the
analyzed samples $\left\\{m_{i}\right\\}$, which describe the sequence of i-th
element partial concentrations in an aerosol, must to obey the Gauss
distribution with respect to the random quantity $\ln{p_{i}}$. Proof of these
assertions for multifractal objects is presented below.

##  4 The spectrum of multifractal dimensions and log normal mass distribution
of secondary aerosol elements

A detailed analysis of Figs. 1, 3-5, where the linear regressions for
different pairs of experimental samples of element concentrations in
atmospheric PM measured at various latitudes are shown, allows us to draw a
definite conclusion about the multifractal nature of PM. The basis of such a
conclusion is the reliably observed linear dependence of Eq. (3) type between
the normalized concentrations $C_{i}$ of the same element i in atmospheric PM
in different regions of the Earth.

Thus, it is necessary to consider the atmospheric PM, which is the
multicomponent (with respect to elements) system, as a nonhomogeneous fractal
object, i.e., as a multifractal. At the same time, the spectrum of fractal
dimensions $f(\alpha)$ and not a single dimension $\alpha_{0}$ (which is equal
to $D_{0}$ for a homogeneous fractal) is necessary for complete description of
a nonhomogeneous fractal object. We will show below that the spectrum of
fractal dimensions $f(\alpha)$ of multifractal predetermines the log normal
type of statistics or, in other words, the log normal type of mass
distribution of multifractal i-th component. This is very important, because
the representation of mass distribution of atmospheric PM as the log normal
distribution is predicted within the framework of the self-preserving theory
ref-20 ; ref-21  and is confirmed by numerous experiments at the same time
ref-3 .

To explain the main idea of derivation we give the basic notions and
definitions of the theory of multifractals. Let us consider a fractal object
which occupies some bounded region £ of size $L$ in Euclidian space of
dimension $d$. At some stage of its construction let it represents the set of
$N>>1$ points distributed somehow in this region. We divide this region £ into
cubic cells of side $\varepsilon<<L$ and volume $\varepsilon^{d}$. We will
take into consideration only the occupied cells, where at least one point is.
Let $i$ be the number of occupied cell $i=1,2,….N(\varepsilon)$, where
$N(\varepsilon)$ is the total number of occupied cells, which depends on the
cell size $\varepsilon$ . Then in the case of a regular (homogeneous) fractal,
according to the definition of fractal dimensions $D$, the total number of
occupied cells $N(\varepsilon)$ at quite small $\varepsilon$ looks like

$N(\varepsilon)\approx\varepsilon_{L}^{-D}=L_{\varepsilon}^{D}.$ (5)

where $\varepsilon$ is the cell size in $L$ units, $L_{\varepsilon}$ is the
size of fractal object in $\varepsilon$ units.

When a fractal is nonhomogeneous a situation becomes more complex, because, as
was noted above, a multifractal is characterized by the spectrum of fractal
dimensions $f(\alpha)$, i.e. by the set of probabilities $p_{i}$, which show
the fractional population of cells $\varepsilon$ by which an initial set is
covered. The less the cell size is, the less its population. For self-
similarity sets the dependence $p_{i}$ on the cell size $\varepsilon$ is the
power function

$p_{i}(\varepsilon)=\frac{1}{N_{i}(\varepsilon)}\approx\varepsilon_{L}^{\alpha_{i}}=L_{\varepsilon}^{-\alpha_{i}}$
(6)

where $\alpha_{i}$ is a certain exponent which, generally speaking, is
different for the different cells $i$. It is obvious, that for a regular
(homogeneous) fractal all the indexes $\alpha_{i}$ in Eq. (6) are identical
and equal to the fractal dimension $D$.

We now pass on to probability distribution of the different values
$\alpha_{i}$. Let $n(\alpha)d\alpha$ is the probability what $\alpha_{i}$ is
in the interval $\left[\alpha,\alpha+d\alpha\right]$. In other words,
$n(\alpha)d\alpha$ is the relative number of the cells $i$, which have the
same measure $p_{i}$ as $\alpha_{i}$ in the interval
$\left[\alpha,\alpha+d\alpha\right]$. According to (5), this number is
proportional to the total number of cells
$N(\varepsilon)\approx\varepsilon_{L}^{-D}$ for a monofractal, since all of
$\alpha_{i}$ are identical and equal to the fractal dimension $D$.

However, this is not true for a multifractal. The different values of
$\alpha_{i}$ occur with probability characterized by the different (depending
on $\alpha$) values of the exponent $f(\alpha)$, and this probability
inherently corresponds to a spectrum of fractal dimensions of the homogeneous
subsets $\text{\textsterling}_{\alpha}$ of initial set £:

$n(\alpha)\approx\varepsilon_{L}^{-f(\alpha)}.$ (7)

Thus, from here a term “multifractal” becomes clear. It can be understood as a
hierarchical joining of the different but homogeneous fractal subsets
$\text{\textsterling}_{\alpha}$ of initial set £, and each of these subsets
has the own value of fractal dimension $f(\alpha)$.

Now we show how the function $f(\alpha)$ predetermines the log normal kind of
mass (volume) distribution of the multifractal $i$-th component. To ease
further description we represent expression (8) in the following equivalent
form:

$n(\alpha)\approx\exp{\left[f(\alpha)\ln{L_{\varepsilon}}\right]}.$ (8)

It is not hard to show ref-19  that the single-mode function $f(\alpha)$ can
be approximated by a parabola near its maximum at the value $\alpha_{0}$.

$f(\alpha)\approx D_{0}-\eta\cdot(\alpha-\alpha_{0})^{2},$ (9)

where the curvature of parabola

$\eta=\frac{f^{\prime\prime}(\alpha_{0})}{2}=\frac{1}{2\left[2(\alpha_{0}-D_{0})+D^{\prime\prime}_{q=0}\right]}$
(10)

is determined by the second derivative of function $f(\alpha)$ at a point
$\alpha_{0}$. Due to a convexity of the function $f(\alpha)$ it is obvious,
that the magnitude in square brackets must be always positive. The fact, that
the last summand $D^{\prime\prime}_{q=0}$ in these brackets is numerically
small and it can be neglected, will be grounded below.

At the large $L_{\varepsilon}$ the distribution $n(\alpha)$ (8) with an
allowance for (9) takes on the form

$n(\alpha)\sim\exp{\left[D_{0}\ln{L_{\varepsilon}}\frac{\left(\alpha-\alpha_{0}\right)^{2}\ln{L_{\varepsilon}}}{4\left(\alpha_{0}-D_{0}\right)}\right]}.$
(11)

Then, taking into account (5), we obtain from (11) the distribution function
of random variable $p_{i}$

$n(p_{i})\sim\ln{N_{D_{0}}}\cdot\exp{\left[-\frac{1}{4\ln{\left(1/p_{0}N_{D_{0}}\right)}}\left(\ln{p_{i}}-\ln{p_{0}}\right)^{2}\right]},$
(12)

which with consideration of normalization takes on the final form

$P(p_{i})=\frac{1}{\sqrt{2\pi}\sigma}\exp{\left(-\mu-\frac{1}{2}\sigma^{2}\right)}\cdot\exp{\left[-\frac{1}{2\sigma^{2}}\left(\ln{p_{i}}-\mu\right)^{2}\right]},$
(13)

where

$\mu=\ln{p_{0}},~{}~{}~{}\sigma^{2}=2\ln{\frac{1}{p_{0}N_{D_{0}}}}.$ (14)

This is the so-called log normal (relative) mass $p_{i}$ distribution. It is
possible to present the first moments of such a kind of distribution for
random variable $p_{i}$ in the following form:

$\left\langle
p\right\rangle=\exp{\left(\mu+\frac{3}{2}\sigma^{2}\right)}=1/p_{0}^{2}N_{D_{0}}^{3}=L_{\varepsilon}^{2\alpha_{0}-3D_{0}},$
(15)

$\mbox{var}(p)=\exp{\left(2\mu\right)}\cdot\left[\exp{\left(4\sigma^{2}\right)}-\exp{\left(3\sigma^{2}\right)}\right]=L_{\varepsilon}^{2\left(2\alpha_{0}-3D_{0}\right)}\left(L_{\varepsilon}^{2\left(2\alpha_{0}-D_{0}\right)}-1\right).$
(16)

At the same time, it is easy to show that the distribution (13) for the random
variable lnpi has the classical Gaussian form

$P\left(\ln{p}\right)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp{\left\\{-\frac{1}{2\sigma^{2}}\left[\ln{p}-\left\langle\ln{p}\right\rangle\right]^{2}\right\\}},$
(17)

where the first moments of this distribution for the random variable
$\ln{p_{i}}$ look like

$\left\langle\ln{p}\right\rangle=\mu+\sigma^{2}=\left(\alpha_{0}-2D_{0}\right)\ln{L_{\varepsilon}},$
(18)

$\mbox{var}\left(\ln{p}\right)=\sigma^{2}=2\left(\alpha_{0}-D_{0}\right)\ln{L_{\varepsilon}}.$
(19)

Thus, according to the known theorem of multidimensional normal distribution
shape ref-22 , a normal law of plane distribution for the two-dimensional
random variable $(p_{1i},p_{2i})$ will be written down as

$P\left(\ln{p_{1i}},\ln{p_{2i}}\right)=\frac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1-r^{2}}}\exp{\left[-\frac{1}{2\left(1-r^{2}\right)}\left(u^{2}+v^{2}-2ruv\right)\right]},$
(20)

where

$u=\frac{\ln{p_{1i}}-\left\langle\ln{p_{1i}}\right\rangle}{\sigma_{1}},~{}~{}~{}v=\frac{\ln{p_{2i}}-\left\langle\ln{p_{2i}}\right\rangle}{\sigma_{2}},$
(21)

$r=\mbox{cov}(\ln{p_{1i}},\ln{p_{2i}})/\sigma_{1}\sigma_{2}$ is the
correlation coefficient between $\ln{p_{1i}}$ and $\ln{p_{2i}}$.

Then, by virtue of well-known linear correlation theorem ref-6 ; ref-22  it is
easy to show that $\ln{p_{1i}}$ and $\ln{p_{2i}}$ are connected by the linear
correlation dependence, if the two-dimensional random variable
$(\ln{p_{1i}},\ln{p_{2i}})$ is normally distributed. This means that the
parameters of two-dimensional normal distribution of the random values
$p_{1i}$ and $p_{2i}$ for the $i$-th component in one aerosol particle, which
are measured in different regions of the Earth (the indexes 1 and 2), are
connected by the equations of direct linear regression:

$\ln{p_{1i}}-\left\langle\ln{p_{1i}}\right\rangle=r\frac{\sigma_{1i}}{\sigma_{2i}}\left[\ln{p_{2i}}-\left\langle\ln{p_{2i}}\right\rangle\right]$
(22)

and inverse linear regression

$\ln{p_{2i}}-\left\langle\ln{p_{2i}}\right\rangle=r\frac{\sigma_{2i}}{\sigma_{1i}}\left[\ln{p_{1i}}-\left\langle\ln{p_{1i}}\right\rangle\right],$
(23)

where $i=1,…,N_{p}$ is the component number.

Taking into consideration that we measure experimentally the total
concentration $C_{i}$ of the $i$-th component in the unit volume of
atmosphere, the partial concentration $m_{i}$ of the $i$-th component in one
aerosol particle measured in different regions of the Earth (the indexes 1 and
2) looks like

$m_{1i}=C_{1i}/n_{1},~{}~{}~{}m_{2i}=C_{2i}/n_{2},$ (24)

where $n_{1}$ and $n_{2}$ are the number of inoculating centers, whose role
play the primary aerosols $(D_{p}<1\mu m)$.

Here it is necessary to make important digression concerning the choice of
quantitative measure for description of fractal structures. According to Feder
ref-18 , determination of appropriate probabilities corresponding to the
chosen measure is the main difficulty. In other words, if choice of measure
determines the search procedure of probabilities $\left\\{p_{i}\right\\}$,
which describe the increment of the chosen measure for given level of
resolution $\varepsilon$, then the probabilities themselves predetermine, in
its turn, the proper method of their measurement. So, general strategy of
quantitative description of fractal objects, in general case, should contain
the following direct or reverse procedure: the choice of measure - the set of
appropriate probabilities - the measuring method of these probabilities.

We choose the reverse procedure. So long as in the present work the averaged
masses of elemental components of atmospheric PM-multifractal are measured,
the geometrical probabilities, which can be constructed by experimental data
for some fixed $\varepsilon$, have the practically unambiguous form:

$p_{i}\left(\varepsilon=const\right)=\frac{m_{i}/\rho_{i}}{\sum\limits_{i}m_{i}/\rho_{i}}=\frac{C_{i}/\rho_{i}}{\sum\limits_{i}C_{i}/\rho_{i}},$
(25)

where $\rho_{i}$ is the specific gravity of the $i$-th component of secondary
aerosol.

Since the random nature of atmospheric PM formation is a priori determined by
the random process of multicomponent diffusion-limited aggregation (DLA), we
used the so-called harmonic measure ref-18  to describe quantitatively a
stochastic surface inhomogeneity or, more precisely, to study an evolution of
possible growth of cluster PM diameter.

In practice, a harmonic measure is estimated in the following way. Because the
perimeter of clusters, which form due to DLA, is proportional to their mass,
the number of knots $N_{p}$ on the perimeter, i.e., the number of possible
growing-points, is proportional to the number of cells $N$ in a cluster. Both
these magnitudes, $N_{p}$ and $N$, change according to the power law (5)
depending on the cluster diameter $L$. From here it follows that all the knots
$N_{p}$, which belong to the perimeter of such clusters, have a nonzero
probability what a randomly wandering particle will turn out in them, i.e.,
they are the carriers of harmonic measure $M_{d}(q,\varepsilon_{L})$:

$M_{d}\left(q,\varepsilon_{L}\right)=\sum\limits_{i=1}^{N_{p}}p_{i}^{q}\cdot\left(\frac{\varepsilon}{L}\right)^{d}=Z\left(q,\varepsilon_{L}\right)\cdot\varepsilon_{L}^{d}\xrightarrow[L\rightarrow\infty]{}\begin{cases}0,&d>\tau(q)\\\
\infty,&d<\tau(q)\end{cases},$ (26)

where $Z(q,L_{\varepsilon})$ is the generalized statistical sum in the
interval $-\infty<q<\infty$, $\tau(q)$ is the index of mass, at which a
measure does not become zero or infinity at
$L\rightarrow\infty(\varepsilon_{L}\rightarrow 0)$.

It is obvious, that in such a form the harmonic measure is described by the
full index sequence $\tau(q)$, which determines according to what power law
the probabilities $p_{i}$ change depending on $L$. At the same time, the
spectrum of fractal dimensions for the harmonic measure is calculated in the
usual way, but using the “Brownian particles-probes” of fixed diameter
$\varepsilon$ for study of possible growth of the cluster diameter $L$. From
(26) it follows that in this case the generalized statistical sum
$Z(q,\varepsilon_{L})$ can be represented in the form

$Z\left(q,\varepsilon_{L}\right)=\sum_{i=1}^{N_{p}}p_{i}^{q}\sim\varepsilon_{L}^{-\tau(q)}.$
(27)

As is known from numerical simulation of a harmonic measure, when the DLA
cluster surface is probed by the large number of randomly wandering particles,
the peaks of “high” asperities in such a fractal aggregate have greater
possibilities than the peaks of “low” asperities. So, if possible growing-
points on the perimeter of our aerosol multifractal to renumber by the index
$i=1,\ldots,N_{p},$ the set of probabilities

$\Re=\left\\{p_{i}\right\\}_{i=1}^{N_{p}},$ (28)

composed of the probabilities of Eq. (25) type will emulate the possible set
of interaction cross-sections between Brownian particle and atmospheric PM-
multifractal surface, which consists of the $N_{p}$ groups of identical atoms
distributed on the surface. Each of these groups characterizes the $i$-th
elemental component in the one atmospheric PM.

A situation is intensified by the fact that by virtue of (17) each of the
independent components obeys the Gauss distribution which, as is known ref-22
, belong to the class of infinitely divisible distributions or, more
specifically, to the class of so-called $\alpha$-stable distributions. This
means that although the Gauss distribution has different parameters (the
average $\mu_{i}=\left\langle\ln{p_{i}}\right\rangle$ and variance
$\sigma_{i}^{2}=\mbox{var}(\ln{p_{i}})$ for each of components, the final
distribution is the Gauss distribution too, but with the parameters
$\mu=\left\langle\ln{p}\right\rangle=\sum\mu_{i}$ and
$\sigma^{2}=\sum\sigma_{i}^{2}$. From here it follows that the parameters of
the two-dimensional normal distribution of all corresponding components in the
plane $\left\\{p_{1},p_{2}\right\\}$ are connected by the equations of direct
linear regression:

$\ln{p_{1}}-\left\langle\ln{p_{1}}\right\rangle=r\frac{\sigma_{1}}{\sigma_{2}}\left[\ln{p_{2}}-\left\langle\ln{p_{2}}\right\rangle\right]$
(29)

and inverse linear regression:

$\ln{p_{2}}-\left\langle\ln{p_{2}}\right\rangle=r\frac{\sigma_{2}}{\sigma_{1}}\left[\ln{p_{1}}-\left\langle\ln{p_{1}}\right\rangle\right].$
(30)

So, validity of the assumption (28) for the secondary aerosol or, that is the
same, validity of the application of harmonic measure for the quantitative
description of aerosol stochastic fractal surface, will be proven if
simultaneous consideration of (28) and equation of direct linear regression
(29) will result in an equation identical to the equation of direct linear
regression (3).

Therefore, taking into account (28) we write down, for example, the equation
of direct linear regression or, in other words, the condition of linear
correlation between the samples of $i$-th component concentrations
$\left\\{\ln{\left(C_{1i}/\rho_{i}\right)}\right\\}$ and
$\left\\{\ln{\left(C_{2i}/\rho_{i}\right)}\right\\}$ in an atmospheric aerosol
measured in different places (indexes 1 and 2):

$\ln{\left(C_{1i}/\rho_{i}\right)}=a_{12}+b_{12}\ln{\left(C_{2i}/\rho_{i}\right)},~{}~{}~{}a_{12}=\frac{1}{N_{p}}\ln{\frac{\prod\limits_{i=1}^{N_{p}}\left(C_{1i}/\rho_{i}\right)}{\left(\prod\limits_{i=1}^{N_{p}}\left(C_{2i}/\rho_{i}\right)\right)^{b_{12}}}},~{}~{}~{}b_{12}=r\frac{\sigma_{1}}{\sigma_{2}}.$
(31)

It is obvious, that this equation completely coincides with the equation of
linear regression (3), but is theoretically obtained on basis of the Gauss
distribution of the random magnitude $\ln{p_{i}}$ and not in an empirical way.

Physical interpretation of the intercept $a_{12}$ is evident from the
expression (29), whereas meaning of the regression coefficient $b_{12}$
becomes clear, according to (19) and (29), from the following expression:

$b_{12}=r\left(\frac{\mbox{var}\left(\ln{p_{1}}\right)}{\mbox{var}\left(\ln{p_{2}}\right)}\right)^{1/2}=r\left[\frac{\ln{\left(L_{1}/\varepsilon_{1}\right)}}{\ln{\left(L_{2}/\varepsilon_{2}\right)}}\right]^{1/2},$
(32)

where $L_{1}$ and $L_{2}$ are the average sizes of separate atmospheric PM-
multifractals typical for the atmosphere of investigated regions (indexes 1
and 2) of the Earth, $\varepsilon_{1}$ and $\varepsilon_{2}$ are the cell
sizes into which the corresponding atmospheric PM-multifractals are divided.

Below we give a computational procedure algorithm for identification of the
generalized fractal dimension $D_{q}$ spectra and function $f(\alpha)$. It is
obvious, that such a problem can be solved by the following redundant system
of nonlinear equations of Eqs. (15), (16), (18) and (19) type:

$\begin{array}[]{c}\ln{\left\langle
p\right\rangle}=\left(2\alpha_{0}-3D_{0}\right)\ln{\left(L/\varepsilon\right)},\\\
\\\
\mbox{var}(p)=\left(L/\varepsilon\right)^{2\left(2\alpha_{0}-3D_{0}\right)}\left[\left(L/\varepsilon\right)^{2\left(2\alpha_{0}-D_{0}\right)}-1\right].\end{array}$
(33)

$\begin{array}[]{c}\left\langle\ln{p}\right\rangle=\mu+\sigma^{2}=\left(\alpha_{0}-2D_{0}\right)\ln{\left(L/\varepsilon\right)},\\\
\\\
\mbox{var}\left(\ln{p}\right)=\sigma^{2}=2\left(\alpha_{0}-D_{0}\right)\ln{\left(L/\varepsilon\right)},\end{array}$
(34)

where $\varepsilon$ is the cubic cell fixed size, into which the bounded
region £ of size $L$ in Euclidian space of dimension $d$ is divided.

To solve the system of equations (33)-(34) with respect to the variables
$\alpha_{0}$, $D_{0}$ and $\varepsilon$, $\varepsilon_{L}$ it is necessary and
sufficient to measure experimentally the $i$-th components of the
concentration sample $\left\\{C_{i}\right\\}$ in the unit volume of
atmospheric air (see section 3) and size distribution of atmospheric PM for
determination of the average size $L$.

It will be recalled that from the physical standpoint so-called the box
counting dimension $D_{0}$, the entropy dimension $D_{1}$ and the correlation
dimension $D_{2}$ are the most interesting in the spectrum of the generalized
fractal dimensions $D_{q}$ corresponding to different multifractal
inhomogeneities. Within the framework of the notions and definitions of
multifractal theory mentioned above we describe below the simple procedure for
finding of spectrum of the generalized fractal dimensions $D_{q}$, taking into
account the solution of the system of equations (33)-(34).

From (27) it follows that in our case a multifractal is characterized by the
nonlinear function $\tau(q)$ of moments $q$

$\tau(q)=\lim_{\varepsilon_{L}\rightarrow
0}\frac{\ln{Z\left(q,\varepsilon_{L}\right)}}{\ln{\varepsilon_{L}}}.$ (35)

As well as before we consider a fractal object, which occupies some bounded
region £ of “running” size $L$ (so that $\varepsilon_{L}\rightarrow 0$) in
Euclidian space of dimension $d$. Then spectrum of the generalized fractal
dimensions $D_{q}$ characterizing the multifractal statistical inhomogeneity
(the distribution of points in the region £) is determined by the relation

$D_{q}=\frac{\tau(q)}{q-1},$ (36)

where $\left(q-1\right)$ is the numerical factor, which normalizes the
function $\tau(q)$ so that the equality $D_{q}=d$ is fulfilled for a set of
constant density in the $d$-dimensional Euclidian space.

Further, we are interested in the known in theory of multifractal connection
between the mass index $\tau(q)$ and the multifractal function $f(\alpha)$ by
which the spectrum of generalized fractal dimensions $D_{q}$ is determined

$D_{q}=\frac{\tau(q)}{q-1}=\frac{1}{q-1}\left[q\cdot
a(q)-f\left(a(q)\right)\right].$ (37)

It is obvious, that in our case, when the sample $\left\\{p_{i}\right\\}$ is
experimentally determined and the cell size $\varepsilon<<L$ is numerically
evaluated (by the system of equations (33)-(34)), the spectrum of generalized
fractal dimensions $D_{q}$ (36) can be obtained by the expression for the mass
index $\tau(q)$ (35):

$D_{q}=\frac{\tau(q)}{q-1}\approx\frac{\ln{\sum\limits_{i=1}^{N_{p}}p_{i}^{q}}}{(q-1)\ln{\varepsilon_{L}}}.$
(38)

Finally, joint using of the Legendre transformation

$\alpha=\frac{d\tau}{dq},$ (39)

$f(\alpha)=q\frac{d\tau}{dq}-\tau,$ (40)

which sets direct algorithm for transition from the variables
$\left\\{q,\tau(q)\right\\}$ to the variables
$\left\\{\alpha,f(\alpha)\right\\}$, and the approximate analytical expression
(38) for the function $D_{q}$ makes it possible to determine an expression for
the multifractal function $f(\alpha)$.

Now we will consider the special case of search of the box counting dimension
$D_{0}$ and the entropy dimension $D_{1}$. One of goals of this consideration
is the validation of assumption of smallness of the magnitude
$D^{\prime\prime}_{q=0}$ in the expression (10), which was used for the
derivation of log normal distribution of the random magnitude $p_{i}$ (13).

It is easy to show that combined using of Eq. (9) and the inverse Legendre
transformation sets, which sets transition from the variables
$\left\\{\alpha,f(\alpha)\right\\}$ to the variables
$\left\\{q,\tau(q)\right\\}$, gives the following dependence of $\alpha(q)$ on
the moments $q$:

$\alpha(q)=2q\left(\alpha_{0}-D_{0}\right)+\alpha_{0}.$ (41)

Substituting (41) and (9) into (37), we obtain the approximate expression for
the spectrum of generalized fractal dimensions $D_{q}(q=0.1)$ depending on
$\alpha_{0}$ and $D_{0}$:

$D_{q}=\frac{1}{q-1}\left[q^{2}\left(\alpha_{0}-D_{0}\right)+q\alpha_{0}-D_{0}\right].$
(42)

Thus, we can write down the expressions for the box counting dimension $D_{0}$
and the entropy dimension $D_{1}$ depending on $\alpha_{0}$ and $D_{0}$

$D_{0}=D_{0},$ (43)

$D_{1}=\lim_{q\rightarrow 1}\frac{q\cdot
a(q)-f\left(a(q)\right)}{q-1}=2D_{0}-\alpha_{0}.$ (44)

Here it is necessary to make a few remarks. It will be recalled that
$f(\alpha_{q=1})=D_{1}$ is the value of fractal dimension of that subset of
the region £, which makes a most contribution to the statistical sum (36) at
$q=1$. However, by virtue of normalizing condition the statistical sum (36) is
equal to unity at $q=1$ and does not depend on the cell size $\varepsilon$, on
which the region £ is divided. Thus, this most contribution also is of order
unity. Therefore, in this case (and only in this case!) the probabilities of
cell occupation $p_{i}\approx\varepsilon_{L}^{\alpha}$ (6) are inversely
proportional to the total number of cells
$n(\varepsilon)\approx\varepsilon_{L}^{-f(\alpha)}$ (7), i.e., the condition
$f(\alpha)=\alpha$ is fulfilled.

So, the parameters of system of the equations (33)-(34) obtained by the
expression (9) can not in essence contain information about the generalized
fractal dimensions $D_{q}$ for absolute value of the moments $q$ greater than
unity (i.e., $|q|>1$).

Secondly, it is easy to show that the expression (39) for the entropy
dimension $D_{1}$ does not depend on concrete type of the function
$f(\alpha)$, but is determined by its properties, for example, by symmetry
$f(\alpha)=\alpha$, $f^{\prime}(\alpha)=1$ and convexity
$f^{\prime\prime}(\alpha)>0$. The geometrical method for determination of the
entropy dimension $D_{1}$ shown in Figure 8 is simultaneously the geometrical
proof of assertion (44).

Figure 8: Geometrical method for determination of the entropy dimension
$D_{1}$, which leads to the obvious equality $D_{1}=2D_{0}-\alpha_{0}$.

Thirdly, the expressions for the entropy dimension $D_{1}$ obtained by
parabolic approximation of the function $f(\alpha)$ and geometrical method
(Figure 8) are equivalent. This means that the magnitude
$D^{\prime\prime}_{q=0}$ in the expression (10) is equal to zero. Thus, our
assumption of smallness of the magnitude $D^{\prime\prime}_{q=0}$ in the
expression (10) is mathematically valid.

In the end, we note that the knowledge of generalized fractal dimensions
$D_{q}$, the correlation dimension $D_{2}$ and especially $D_{1}$, which
describes an information loss rate during multifractal dynamic evolution,
plays the key role for an understanding of the mechanism of secondary aerosol
formation, since makes it possible to simulate a scaling structure of an
atmospheric PM with well-defined typical scales. Returning to the initial
problem of distribution of points over the fractal set £, it is possible to
say that the magnitude $D_{1}$ gives an information necessary for
determination of point location in some cell, while the correlation dimension
$D_{2}$ determines the probability what a distance between the two randomly
chosen points is less than $\varepsilon_{L}$. In other words, when the
relative cell size tends to zero ($\varepsilon_{L}\rightarrow 0$), these
magnitudes are anticorrelated, i.e. the entropy $D_{1}$ decreases, while the
multifractal correlation function $D_{2}$ increases.

##  5 Conclusions

Comparative analysis of different pairs of experimental normalized
concentration values of atmospheric PM elements measured in different regions
of the Earth shows a stable linear (on a logarithmic scale) correlation
($r=1$) dependence on different time scales (from average daily to annual).
That points to a power law increase of every atmospheric PM element mass
(volume) and simultaneously to the cause of this increase – the fractal nature
of the genesis of atmospheric PM.

Within the framework of multifractal geometry it is shown that the mass
(volume) distribution of the atmospheric PM elemental components is a log
normal distribution, which on the logarithmic scale with respect to the random
variable (elemental component mass) is identical to the normal distribution.
This means that the parameters of the two-dimensional normal distribution with
respect to corresponding atmospheric PM-multifractal elemental components,
which are measured in different regions, are a priory connected by equations
of direct and inverse linear regression, and the experimental manifestation of
this fact is the linear (on a logarithmic scale) correlation between the
concentrations of the same elemental components in different sets of
experimental atmospheric PM data.

We would like to note here that a degree of our understanding of the mechanism
of atmospheric PM formation, which due to aggregation on inoculating centres
(primary aerosols ($D_{p}<1\mu m$)) show a scaling structure with well-defined
typical scales, can be described by the known phrase: “ …we do not know till
now why clusters become fractals, however we begin to understand how their
fractal structure is realized, and how their fractal dimension is related to
the physical process” ref-23 . This made it possible to show that the spectrum
of fractal dimensions of multifractal, which is a multicomponent (by elements)
aerosol, always predetermines the log normal type of statistics or, in other
words, the log normal type of mass (volume) distribution of the i-th component
of atmospheric PM.

It is theoretically shown, how solving the system of nonlinear equations
composed of the first moments (the average and variance) of a log normal and
normal distributions, it is possible to determine the multifractal function
$f(\alpha)$ and spectrum of fractal dimensions $D_{q}$ for separate averaged
atmospheric PM, which are the global characteristics of genesis of atmospheric
PM and does not depend on the local place of registration (measurement).

We should note here that the results of this work allow an approach to
formulation of the very important problem of aerosol dynamics and its
implications for global aerosol climatology, which is connected with the
global atmospheric circulation and the life cycle of troposphere aerosols
ref-3 ; ref-21 . It is known that absorption by the Earth’s solar short-wave
radiation at the given point is not compensated by outgoing long-wave
radiation, although the integral heat balance is constant. This constant is
supported by transfer of excess tropical heat energy to high-latitude regions
by the aid of natural oceanic and atmospheric transport, which provides the
stable heat regime of the Earth. It is evident that using data about elemental
and dispersed atmospheric PM composition in different regions of the Earth
which are “broader-based” than today, one can create the map of latitude
atmospheric PM mass and size distribution. This would allow an analysis of the
interconnection between processes of ocean-atmosphere circulation and
atmospheric PM genesis through the surprising ability of atmospheric PM for
long range transfer, in spite of its short “lifetime” (about 10 days) in the
troposphere. If also to take into consideration the evident possibility of
determination of latitude inoculating centers (i.e., primary aerosol)
distribution, this can lead to a deeper understanding of the details of
aerosol formation and evolution, since the natural heat and dynamic
oscillations of the global ocean and atmosphere are quite significant and
should impact influence primary aerosol formation dynamics and fractal genesis
of secondary atmospheric aerosol, respectively.

It is important to note also that continuous monitoring of the main
characteristics of South Pole aerosols as a standard of relatively pure air,
and the aerosols of large cities, which are powerful sources of anthropogenic
pollution, allows to determine the change of chemical and dispersed
compositions of aerosol pollution. Such data are necessary for a
scientifically-founded health evaluation of environmental quality, as well as
for the planning and development of an air pollution decrease strategy in
cities.

## References

  * (1) Maenhaut, W.; Zoller, W.H. Determination of the chemical composition of the South Pole aerosol by instrumental neutron activation analysis. J. Radioanal. Chem. 1977, 37, 637-650.
  * (2) Pushkin, S.G.; Mihaylov, V.A. Comparative Neutron Activation Analysis: Study of Atmospheric Aerosols; Nauka, Siberian Department: Novosibirsk, 1989.
  * (3) Raes, F.; van Dingenen, R.; Vignati, E.; Wilson, J.; Putaud, J.P.; Seinfeld J.H.; Adams, P. Formation and cycling of aerosols in the global troposphere. Atmos. Environ. 2000, 34, 4215-4240.
  * (4) Rusov, V.D.; Glushkov, A.V.; Vaschenko, V.N. Astrophysical Model of the Earth Global Climate; Naukova Dumka: Kiev, 2003 (in Russian).
  * (5) Figen, Var; Yasushi Narita; Shigeru Tanaka. The concentration, trend and seasonal variation of metals in the atmosphere in 16 Japanese cities shown by the results of National Air Surveillance Network (NASN) from 1974 to 1996. Atmos. Environ. 2000, 34, 2755-2770.
  * (6) Brownlee, K.A. Statistical Theory and Methodology in Science and Engineering; Ed. John Wiley & Sons: New York, 1965.
  * (7) Bendat, J.S.; Piersol, A.G. Random Data: Analysis and Measurement Procedures; Ed. John Wiley & Sons: New York, 1986.
  * (8) Schroeder, M. Fractals, Chaos, Power Laws: Minutes from Infinite Paradise; Ed. W. Freeman and Company: New York, 2000.
  * (9) Witten, T.A.; Sander, L.M. Diffusion-limited aggregation: kinetic critical phenomenon. Phys. Rev. Lett.1981, 47, 1400-1403.
  * (10) Zubarev, A.Yu.; Ivanov, A.O. Fractal structure of colloid aggregate. Reports of Russian Academy of Sci. 2002, 383, 472-477.
  * (11) Maenhaut, W.; Francos, F.; Cafmeyer, J. The “Gent” Stacked Filter Unit (SFU) Sampler for Collection of Atmospheric Aerosols in Two Size Fractions: Description and Instructions for Installation and Use. Report No.NAHRES-19, IAEA: Vienna, 1993, pp. 249-263.
  * (12) Hopke, P.K.; Hie, Y.; Raunemaa, T.; Biegalski, S.; Landsberger, S.; Maenhaut, W.; Artaxo, P.; Cohen, D. Characterization of the Gent stacked filter unit $PM_{10}$ Sampler. Aerosol Sci. Tech. 1997, 27, 726-735.
  * (13) Jaćimović, R.; Lazaru, A.; Mihajlović, D;, Ilić, R;, Stafilov, T. Determination of major and trace elements in some minerals by $k_{0}$-instrumental neutron activation analysis. J. Radioanal. Nucl. Chem. 2002, 253, 427-434.
  * (14) HYPERMET-PC V5.0, User’s Manual; Institute of Isotopes: Budapest, Hungary, 1997.
  * (15) Kayzero/Solcoi® ver. 5a. User’s Manual for Reactor-neutron Activation Analysis (NAA) Using the $k_{0}$-Standardization Method; DSM Research: Geleen, Netherlands, 2003.
  * (16) Cronover, R.M. Introduction to Fractals and Chaos; Jones and Bartlett Publishers, 1995.
  * (17) Mandelbrot, B.B. The fractal geometry of nature. Updated and Augmented; W.H. Freeman and Company: New York, 2002.
  * (18) Feder, J. Fractals; Plenum Press: New York, 1988.
  * (19) Bozhokin, S.V.; Parshin, D.A. Fractals and Multifractals; Scientific Publishing Centre "Regular and Chaotic Dynamics": Moscow-Izhevsk, 2001 (in Russian).
  * (20) Lai, F.S.; Friedlander, S.K.; Pich, J.; Hidy, G.M. The self-preserving particle size distribution for Brownian coagulation in the free-molecular regime. J. Colloid Interf. Sci. 1972, 39, 395-405.
  * (21) Raes, F.; Wilson, J.; van Dingenen, R. Aerosol dynamics and its implication for the global aerosol climatology. In Aerosol Forcing of Climate; Charson, R.J., Heintzenberg, J., Eds.; John Wiley & Sons: New York, 1995.
  * (22) Feller, W. An Introduction to Probability Theory and its Applications; John Wiley & Sons: New York, 1971.
  * (23) Bote, R.; Julen, R.; Kolb, M. Aggregation of Clusters. In Fractals in Physics; Pietronero, L., Tosatti, E., Eds.; North-Holland :Amsterdam, 1986, pp. 353-359.


An Adaptive Optimization Method for District Heating
Networks]Adaptive Nonlinear Optimization of District Heating Networks
Based on Model and Discretization Catalogs

H. Dänschel]Hannes Dänschel
V. Mehrmann]Volker Mehrmann
M. Roland]Marius Roland
M. Schmidt]Martin Schmidt

[H. Dänschel, V. Mehrmann]
TU Berlin, Inst. f. Mathematik, MA 4-5,
Str. des 17. Juni 135, 10623 Berlin, Germany

[M. Roland, M. Schmidt]Trier University,
Department of Mathematics,
Universitätsring 15,
54296 Trier,

We propose an adaptive optimization algorithm for operating district
heating networks in a stationary regime.
The behavior of hot water flow in the pipe network is modeled using
the incompressible Euler equations and a suitably chosen energy
By applying different simplifications to these equations, we derive a
catalog of models.
Our algorithm is based on this catalog and adaptively controls
where in the network which model is used.
Moreover, the granularity of the applied discretization is controlled
in a similar adaptive manner.
By doing so, we are able to obtain optimal solutions at low
computational costs that satisfy a prescribed tolerance w.r.t. the
most accurate modeling level.
To adaptively control the switching between different levels and the
adaptation of the discretization grids, we derive error measure formulas
and a posteriori error measure estimators.
Under reasonable assumptions we prove that the adaptive algorithm
terminates after finitely many iterations.
Our numerical results show that the algorithm is able to produce
solutions for problem instances that have not been solvable before.

journaltitleEuropean Journal of Operational Research
titleAn optimization approach for district heating strategic network design

titleA local time stepping method for district heating networks

Springer, Heidelberg

titleHandbook of Mathematics

noteThe 24th International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy, ECOS 2011
titleNumerical analysis of a medium scale latent energy storage unit for district heating systems

noteTRR 154 Preprint
titleGas Network Modeling: An Overview

journaltitleIEEE Transactions on Smart Grid
titleLarge-Scale District Heating Network Optimization

journaltitleORSA Journal on computing
titleCONOPT—a large-scale GRG code

journaltitleEURO Journal on Computational Optimization
titleComplementarity-based nonlinear programming techniques for optimal mixing in gas networks

titlePyomo-optimization modeling in Python

journaltitleMathematical Programming Computation
titlePyomo: modeling and solving mathematical programs in Python

journaltitleApplied Thermal Engineering
noteCombined Special Issues: ECP 2011 and IMPRES 2010
titleImpact of load structure variation and solar thermal energy integration on an existing district heating network

booktitleProgress in Differential Algebraic Equations II
seriesDifferential-Albergaic Equations Forum
titlePort-Hamiltonian modeling of district heating networks

Amsterdam: North Holland

titlePerturbation Theory for Matrix Equations
pagesxii + 429

journaltitleOptimization and Engineering
titleNonlinear Optimization of District Heating Networks

booktitle58th IEEE Conference on Decision and Control (CDC), 9.-12.12.19, Nice
titleStructure-preserving discretization for port-Hamiltonian descriptor systems

journaltitleVietnam Journal of Mathematics
titleModel and Discretization Error Adaptivity Within Stationary Gas Transport Optimization

booktitleMultiscale, nonlinear and adaptive approximation
titleTheory of adaptive finite element methods: an introduction

titleInfluence of system design on heat distribution costs in district heating
District heating,Linear heat density,Connection load,Optimum pipe diameter

titleEnergy consumption and economic analyses of a district heating network

Springer Science & Business Media

titleNumerical Mathematics

seriesMathematics in Science and Engineering
titleRiccati Differential Equations

titleModel order reduction of hyperbolic systems at the example of district heating networks

titleOptimal control of district heating networks using a reduced order model

titleParametric model order reduction for district heating networks

journaltitleApplied Energy
titleDistrict heating and cooling: Review of technology and potential enhancements

journaltitleat - Automatisierungstechnik
noteSpecial Issue "Mathematical Innovations fostering the Energy Transition – Control, Optimization and Uncertainty Quantification"
titleMixed-Integer Nonlinear Optimization for District Heating Network Expansion

booktitleProceedings of the 44th IEEE Conference on Decision and Control
titlePredictive Control of a Complex District Heating Network
Predictive control;Heating;Temperature control;Robust control;Optimal control;Costs;Power generation economics;Economic forecasting;Uncertainty;Production

journaltitleOptimization and Engineering
titleHigh detail stationary optimization models for gas networks: validation and results

titleDistrict heating and cooling systems – Framework for Modelica-based simulation and dynamic optimization

Springer, New York, NY

titleIntroduction to Numerical Analysis

journaltitleAnalele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica
titleError Analysis and Model Adaptivity for Flows in Gas Networks

titlePrimary energy savings through thermal storage in district heating networks

OUP Oxford

titleA posteriori error estimation techniques for finite element methods
family=Del Vecchio,

journaltitleIEEE Transactions on Automation Science and Engineering
titleModel Predictive Control-Based Optimal Operations of District Heating System With Thermal Energy Storage and Flexible Loads