Estimation of Radon Flux Density Changes in Temporal Vicinity of the Shipunskoe Earthquake with Mw = 7.0, 17 August 2024 with the Use of the Hereditary Mathematical Model

Abstract

Using the data of radon accumulation in a chamber with excess volume at one of the points of the Kamchatka subsurface gas-monitoring network, the change in radon flux density due to seismic waves and post-seismic relaxation of the medium is shown. A linear fractional equation is considered to be a model equation. The change of radon-transport intensity due to changes in the state of the geo-environment is described by a fractional Gerasimov–Caputo derivative of constant order. Presumably, the order of the fractional derivative is related to the radon-transport intensity in the geosphere. Using the Levenberg–Marquardt method, the optimal values of the model parameters were determined based on experimental data: air exchange coefficient and order of fractional derivative, which allowed the solving of the problems of radon flux density determination. Data in the temporal neighborhood of a strong earthquake with Mw=7.0, which occurred in the northern part of Avacha Bay on 17 August 2024, were used. As a result of the modeling, it is shown that the strong seismic impact and subsequent processes led to changes in the radon flux in the accumulation chamber. The obtained model curves agree well with the real data, and the obtained estimates of radon flux density agree with the theory.

Full text

Academic Editor: Dimitrios
Nikolopoulos
Received: 30 November 2024
Revised: 9 January 2025
Accepted: 14 January 2025
Published: 16 January 2025
Citation: Tverdyi, D.; Makarov, E.;
Parovik, R. Estimation of Radon Flux
Density Changes in Temporal Vicinity
of the Shipunskoe Earthquake with
Mw=7.0, 17 August 2024 with the
Use of the Hereditary Mathematical
Model. Geosciences 2025,15, 30.
https://doi.org/10.3390/
geosciences15010030
Copyright: © 2025 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license
(https://creativecommons.org/
licenses/by/4.0/).
Article
Estimation of Radon Flux Density Changes in Temporal Vicinity
of the Shipunskoe Earthquake with Mw=7.0, 17 August 2024
with the Use of the Hereditary Mathematical Model
Dmitrii Tverdyi 1, Evgeny Makarov 2and Roman Parovik 1,*
1Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Mirnaya Street 7,
684034 Kamchatka, Russia; tverdyi@ikir.ru
2Kamchatka Branch of the Federal Research Center “Unified Geophysical Service of the Russian Academy of
Sciences”, Petropavlovsk-Kamchatsky, Piipa Boulevard St. 9, 683023 Kamchatka, Russia; ice@emsd.ru
*Correspondence: parovik@ikir.ru
Abstract: Using the data of radon accumulation in a chamber with excess volume at
one of the points of the Kamchatka subsurface gas-monitoring network, the change in
radon flux density due to seismic waves and post-seismic relaxation of the medium is
shown. A linear fractional equation is considered to be a model equation. The change of
radon-transport intensity due to changes in the state of the geo-environment is described
by a fractional Gerasimov–Caputo derivative of constant order. Presumably, the order
of the fractional derivative is related to the radon-transport intensity in the geosphere.
Using the Levenberg–Marquardt method, the optimal values of the model parameters were
determined based on experimental data: air exchange coefficient and order of fractional
derivative, which allowed the solving of the problems of radon flux density determination.
Data in the temporal neighborhood of a strong earthquake with
Mw=
7.0, which occurred
in the northern part of Avacha Bay on 17 August 2024, were used. As a result of the
modeling, it is shown that the strong seismic impact and subsequent processes led to
changes in the radon flux in the accumulation chamber. The obtained model curves
agree well with the real data, and the obtained estimates of radon flux density agree with
the theory.
Keywords: mathematical modeling; radon volumetric activity; radon flux density;
earthquake
precursors; Gerasimov–Caputo fractional derivative; memory effect;
non-locality
; inverse
problems; Levenberg–Marquardt algorithm; MATLAB; C; Gnuplot
MSC: 26A33; 86A22; 49N45
1. Introduction
Throughout the geologic history of Earth, water–gas fluid flows have been continu-
ously emitted from its interior, which vary markedly in time and are unevenly distributed
on the surface of our planet, reflecting its geodynamic regime, block structure, and place-
ment of fault zones [
1
,
2
]. At present, geophysics has developed the idea of geological
medium as a hierarchically self-similar open non-equilibrium system of separates, in which
the seismic process is a consequence of the deformation of the medium under the action
of tectonic forces. The potential energy accumulated in the structural volumes of such
medium during its deformation is dissipated mainly on the systems of faults and blocks
of different scale levels. As modern experimental and theoretical studies have shown, the
diagnosis of criticality level and prediction of catastrophic earthquakes should be based,
Geosciences 2025,15, 30 https://doi.org/10.3390/geosciences15010030

Geosciences 2025,15, 30 2 of 23
first, on the existing interaction between global and local geomechanical fields in the upper
part of Earth’s crust of block-hierarchical structure.
The recorded character and peculiarities of the response of the medium to earthquake
preparation depend on the type of observations, the type of equipment used, and the
strain sensitivity of the observation point. Works on investigation of the connection of
radon gas dynamics (
222Rn
) with seismicity are carried out in many countries located
in seismically active areas of the world (Israel, India, Japan, USA, China, Russia). The
precursor anomalies are sought in the dynamics of subsurface radon, radon in atmospheric
air, and dissolved radon in thermal waters. Examples of such precursor anomalies have
been published in numerous articles, a review of which can be found in the works of [
3
–
8
],
where numerous data on radon precursors are collected and systematized. The paper [
9
]
is the most relevant and contains a large literature review of electromagnetic and radon
precursors for earthquakes. This work also details mathematical methods for analyzing
data from electromagnetic radiation and radon concentration measurements and explores
physical models of earthquake occurrence to interpret the causes of precursors [9].
As a rule, the literature considers the amplitude of the anomaly, the lead time to the
moment of seismic event occurrence, and the duration of the anomaly. In almost all works
devoted to radon precursors, their selection is based on the temporal confinement to strong
earthquakes relative to the background, as a rule, in one point [
10
–
15
]. Due to the rapid
progress of observational tools, automatic stations are now being used to comprehensively
record the gas composition of subsurface air [
16
]. In this case, anomalies are highlighted
with greater certainty. The lead times of radon earthquake precursors range from hours to
several years, i.e., both short-term and long-term precursors are observed. Therefore, the
occurrence of precursors with such a time range of anticipation is clearly due to various
geodynamic processes that require their own study. The amplitude of anomalies is most
often within 20–200% of the background, but cases of exceeding by more than 1200%
have been noted. It should be noted that negative anomalies were also observed among
radon precursors.
The prospect of the radon method for the purpose of monitoring geodynamic pro-
cesses, in particular, predicting earthquakes and mountain shocks, has been shown in
numerous works, references to which can be found in reviews [
4
–
6
,
9
,
17
]. These works
describe various possible mechanisms of radon anomalies related to changes in fracturing
and permeability of the medium, changes in fluid flow, etc. Work is underway to monitor
subsurface gas concentrations in order to predict volcanic eruptions [
18
,
19
]. A number
of review papers, where numerous data on radon precursors have been collected and
systematized, describe anomalies with lead times ranging from hours to several years.
Thus, in the work of more than 30 years old [
3
] based on data for a ten-year observation
period in the territory of China, 4 types of radon precursors with different lead times were
identified: long-term (several years), medium-term (about a year), short-term
(2–6 months),
and operational (hours to days). In the paper [
4
], the analysis of 83 known at that time
radon precursors registered in different seismically active regions of the world by the
end of the 80s of the last century was made: 28 in Central Asia on the territory of the
USSR, 15 in China, 4 in the Caucasus, 32 in North America (USA). Attention was paid to
their temporal forms, and empirical relationships were found between the parameters of
precursor anomalies (amplitude, preemptive time) and earthquake parameters (magnitude,
distance). In a review work done ten years later [
5
], attention is drawn to the peculiarities
of radon anomalies, and an attempt is made to explain their physical nature. The work,
written about a quarter of a century ago, and today has not lost its relevance. It notes that
radon precursors have a great variety of forms of different duration and are registered at
considerable distances from the epicenters of both shallow and deep earthquakes with a

Geosciences 2025,15, 30 3 of 23
range of magnitudes
Mw=
4
−
8. In the opinion of the author of [
5
], the observed anoma-
lies
222Rn
may occur at small deformations associated with changes in the stress-strain
state of the geosphere at the point of observation. In the paper [
6
], the analysis of radon
precursors is carried out on a wider material, data on 125 precursors in the radon field
before 86 earthquakes with
Mw=
2.5
−
8 are given. Most observations were made before
earthquakes with Mw>4.
Modern researchers continue to consider issues related to the influence of deformation
processes on radon migration in rocks and various materials [
20
–
22
]. In [
23
], it is shown
that at triaxial compression of rock salt formation, the emanation of geogenic noble gases
correlates with volumetric strain and acoustic emission. At low pressures, the rock salt
deforms mainly due to the destruction of crystalline grains and the emanation of gases
with a large amount of acoustic emission. At higher pressures, the number of crystalline
grains that collapse decreases, and it is assumed that rock salt deforms plastically, emitting
fewer gases with less acoustic emission. The authors show that geogenic gas emission
during deformation can provide additional clues containing information about the type
and magnitude of deformation occurring in Earth’s crust [23].
One of the tasks of modern geodynamics related to seismic activity and generation
of
222Rn
volume activity anomalies is to identify the physical mechanisms of propagation
and redistribution of energy of deformation processes in the geological medium. The
block structure of Earth’s crust and lithosphere significantly affects deformation, seismic,
filtration, and other [
24
,
25
] processes. The correlation between seismic and radon activity is
confirmed by numerous studies. It shows that seismicity and generation of RVA anomalies
are controlled by some internal process and serve as its manifestation on Earth’s surface.
As follows from the brief review above, the potential of joint radon and deformation
monitoring for theoretical understanding of the mechanisms of radon concentration varia-
tions and their relation to tectonics and earthquake preparation, as well as the prediction of
strong earthquakes, is beyond doubt.
Seismic activity in the modern world is one of the most dangerous threats to human life.
Examples of such earthquakes that have occurred in the 21st century are the earthquakes
in Turkey and Syria (6 February 2023,
Mw=
7.7, which killed about 5000 people), the
earthquake off the east coast of Honshu Island in Japan (11 March 2011,
Mw=
9.1, which
caused a catastrophic tsunami, the largest man-made disaster at a nuclear power plant, and
killed about 20,000 people), the earthquake in the Indian Ocean on 26 December 2004 with
a magnitude from
Mw=
9.1 to
Mw=
9.3. The focal point (hypocenter) was located 160
km to the west of Sumatra Island (Indonesia), at a depth of 30 km. The resulting tsunami,
up to 10 m high, reached the coasts of Indonesia, Thailand, Sri Lanka, the south of India,
and the east coast of Africa. As a result, it killed, according to various estimates, 225,000 to
300,000 people in 14 countries and affected about 2.2 million people. The earthquake and
tsunami caused widespread destruction. The economic loss of Thailand was estimated at
USD 5 billion, Indonesia USD 4.5 billion, India USD 4.5 billion, Sri Lanka USD 1.6 billion,
and Maldives USD 1.3 billion. This is by no means a complete list of seismic catastrophes
of the 21st century, but it already demonstrates the danger that Earth’s depths conceal
in themselves.
For the Kamchatka region, the problem of seismic activity has the highest priority,
as the Kamchatka peninsula is one of the most earthquake-prone areas on Earth. The
cities of Petropavlovsk-Kamchatsky, Yelizovo, and Vilyuchinsk are in the zone of impact of
shaking of the highest category (up to 9 units), which once again proves the need to develop
effective methods of seismic activity forecasting. A possible way to solve this problem is to
create an effective forecasting system based on an in-depth understanding of the processes
occurring in Earth’s crust in preparation for a future earthquake source. One of the methods

Geosciences 2025,15, 30 4 of 23
of studying such processes is the monitoring of changes in the concentration of radon gas
(
222Rn
) in the subsurface, atmospheric air, and water [
26
]. Such changes may serve as
precursor signals for the preparation of strong earthquakes. The study of 222Rn variations
in subsurface air helps to obtain information about geodynamic processes in the mountain
massif and properties of the medium through which subsurface gases migrate. Long-term
observations and their comparison with the seismic activity of the region make it possible
to track changes in the character of geodynamic processes. This information can be used to
search for precursor anomalies that may indicate the approach of a strong earthquake.
One of the uranium decay products is the noble gas
222Rn
. The decay of radon and
its products occurs with the release of three types of radiation, by which its concentration
can be measured. By diffusion and convection, radon is transported into the atmosphere.
The Petropavlovsk-Kamchatsky geodynamic test site monitors 222Rn at a network of sites.
The network is created in such a way that the main attention is directed to the Avacha and
Kronotsky Bays and the south of Kamchatka. The observation points are equipped with
complexes for recording subsurface gases, primarily radon and hydrogen. The network
has been operating in different configurations for more than 20 years. Based on the data
obtained, including radon volumetric activity (RVA), two types of precursor anomalies
were identified in the subsurface radon field for a number of earthquakes in the Avacha Gulf
region. Type A anomalies are characterized by their in-phase manifestation at 3–5 points
with a relative shift in time. The most common forms of these anomalies are bay-shaped
and stepped, of different polarity. The precursor anomaly of type A was also registered
for the deep (origin depth 177 km) Zhupanovskoe earthquake with
Mw=
7.2 and for the
earthquake on 3 April 2023 with
Mw=
6.9. The possible cause of in-phase anomalies on
the network of points of registration of subsurface gases is deformation processes such as a
solitary “deformation wave”, which can arise due to the quasi-viscous flow of geomaterial
at the last stage of earthquake preparation. Type B precursor anomalies were registered, as
a rule, in one point of the network, had a definite shape and were well distinguished on the
general background.
Calculations performed by mathematical modeling methods have shown that the
mechanism of such anomalies is associated with the injection of
222Rn
into the ground-
water flow with complete transverse mixing under the influence of a deformation stress
pulse [
27
]. Identification of signs of such processes in radon observation data is one of
the urgent tasks of geophysics. Application of new experimental data of subsurface gas
variations in comparison with data on the seismic activity of the region and investigation
of
222Rn
migration processes with the help of the most modern methods of mathematical
modeling in order to interpret anomalous effects preceding earthquakes is a new and actual
research method in Earth sciences and, in particular, in the development of earthquake
prediction methods.
An earthquake with magnitude
Mw=
7.0 happened on 17 August 2024 at 19:10:26
(UTC) in the northern part of the Avacha Gulf, not far from the Shipunsky peninsula. This
earthquake was later named Shipunsky in connection with the geographical location of the
epicenter (52.931° N, 160.133° E). The epicentral distance to Petropavlovsk-Kamchatsky was
100 km, and the depth was 29 km, according to the National Earthquake Information Center
(NEIC, U.S. Geological Survey). The intensity of shaking in Petropavlovsk-Kamchatsky
was 6–7 points on the seismic intensity scale (SIS). Before this earthquake, a number of
anomalous effects were registered in the subsurface radon field at the network of points [
7
]
near Petropavlovsk-Kamchatsky. The data from radon monitoring before and after this
earthquake were used in this work to estimate the changes in the radon flux density in the
accumulation chamber.

Geosciences 2025,15, 30 5 of 23
The article has the following structure. Section 1presents the introduction, literature
review, and structure of the paper. Section 2describes the methodology of the study,
specifically the emanation method and the fractional derivative method. Section 3briefly
summarizes radon monitoring as the main method for obtaining the experimental RVA
data used in the study. Section 4describes the method for solving the direct problem for the
hereditary
α
-model RVA. Section 5formulates the inverse problem of recovering the values
of several constant parameters
λ0
and
α
and describes the method of its solution. Section 6
introduces the concept of radon flux density and its relation to seismic activity and describes
the method of radon flux density estimation. Section 7presents the results of solving the
inverse problems for
λ0
and
α
on the basis of different experimental RVA data, estimates
the radon flux density on the basis of the reconstructed values of the parameters of the
hereditary
α
-model RVA, and analyzes the sensitivity of the solution of the inverse problem.
Section 8summarizes the results of the study, formulates conclusions, and indicates further
possible research direction.
2. Methodology
The research methodology in the article is based on the theory of the emanation
method and the method of fractional derivatives. Let us consider them in more detail.
2.1. Emanation Method
The theory of the emanation method is based on the study of the process of mass
transfer of radioactive gases (emanations) in the geological medium, as well as their
flow into the surface layer of the atmosphere. The theory of the emanation method was
developed for the purpose of searching for minerals (uranium ores) in the 1930s, but
today, it is actively used to solve various problems, for example, related to the transfer of
radionuclides the atmosphere [
28
], as well as in the tasks of prediction of strong earthquakes
and mining strikes in deep mines [
9
,
29
,
30
]. The radioactive inert gas radon (
222Rn
), which
is the decay product of radium (
226Ra
), is usually considered to be an emanation (Figure 1).
Figure 1. The radioactive series 238 U.
The research instrument in the scope of the theory of the emanation method is mathe-
matical modeling. Mathematical modeling allows the development of various models of the
radon-transport process in geo-medium. In diffusion-convective models, the mechanism of

Geosciences 2025,15, 30 6 of 23
radon transport occurs due to diffusion and convection (advection) [
31
]. In such models,
radon transport is carried out along the spatial coordinate (stationary transport) as well as
along time (non-stationary transport). In models of radon accumulation in chambers or
confined spaces (mines), an important role is played by radioactive decay, radon entry, and
exit through cracks and crevices [
32
]. In this article, we will consider the process of radon
accumulation in a chamber according to the model [32]:
dA(t)
dt =S(t)−A(t)−Aatm
Rn λv(t)−A(t)−Aatm
Rn λRn ,A(t0) = A0, (1)
where
A(t)
—radon volumetric activity (RVA), [Bq/m
3
];
Aatm
Rn
—RVA in outside air, [Bq/m
3
];
S(t)
—function responsible for the
222Rn
entering the chamber, [Bq/m
3
s];
λv(t)
—function
responsible for the air exchange in the chamber, [s
−1
];
λRn =
2.1
·
10
−6
—radon decay con-
stant, [s
−1
];
t∈[t0
,
T]
—time of the process under consideration, [s];
t0and T>0—initial
and final time moments, [s];
Remark 1. The model (1) may allow some simplifications. For example, the third term in the
right-hand side of Equation (1) can be neglected because
λv(t)≫λRn
. This is due to the fact that
even in a completely closed room, the values of the air exchange coefficient are an order of magnitude
larger than the decay constant λRn.
Remark 2. It is known that the transport of
222Rn
in the vertical direction can be caused by
thermofluidic convection, turbulent effects due to changes in meteorological factors, diffusion due
to the pressure gradient in Earth’s crust, diffusion due to the concentration gradient of
222Rn
, and
others [
5
]. Therefore, in the model (1), the value
S(t)
describes simultaneously the convective
SC(t)
and diffusive
S∆(t)
mechanisms. Since there is no significant convective flow of subsurface air from
the surface under the accumulation chamber, the term SC(t)can be ignored.
Remark 3. The modeled accumulation process
222Rn
can be considered stationary in the sense
of inflow rate
222Rn
when the radon flux density (RFD) from the surface under the accumulation
chamber is constant and when there are no sudden changes in the air exchange coefficient. Then
the model parameters
λv(t) = λ0
and
S(t) = S
are constant values, and the RVA will have
an accumulative character with saturation yield:
Amax =S/λ0
[
7
]. Whence we obtain that
S=Amax λ0.
Taking into account the remarks, the model (1) can be rewritten in a simplified form
as follows: dA(t)
dt =λ0(Amax −A(t)),A(t0) = A0. (2)
Let us rewrite Equation (2) for RVA in dimensionless form by normalizing by Am ax :
d¯
A(t)
dt =λ0(1−¯
A(t)),¯
A(t0) = ¯
A0,¯
A(t) = A(t)
Amax ,¯
A0=A0
Amax . (3)
The solution of the model (3) is known, and it is of the form:
¯
A(t) = 1−e−λ0t(1−¯
A0),¯
A(t0) = ¯
A0. (4)
The solution (4) describes the accumulation process
222Rn
in the chamber and has an
asymptotics ¯
A(t)→1 at t→∞(Figure 2).

Geosciences 2025,15, 30 7 of 23
Figure 2. Solution graph (4).
The solution (4) is often applied to investigate the dynamics of RVA in an accumulation
chamber when the geo- medium experiences a stress-strain state [7].
2.2. Fractional Derivative Method
The theory of the emanation method is based on its research on the fundamental prin-
ciples of classical thermodynamics of non-reversible processes. Some of these principles,
the principle of locality and the principle of local thermodynamic equilibrium, significantly
impose limitations on the applications of this theory. This is because the process of mass
transfer is nonlocal and occurs only in those systems that are not in a state of thermodynamic
equilibrium. Therefore, there are often cases where radon has a high migration capacity [
33
,
34
].
It is impossible to explain this using only the diffusion mechanism. Diffusion does not give
radon a chance to move to a distance of the order of more than ten meters before the decay
of radon atoms reduces its concentration to a level indistinguishable from the background,
even if its source is strong enough. Convection (advection) can also contribute to radon
migration ability, but it can also be insignificant. Therefore, it is necessary to search for
other mechanisms of transport or to study the properties of the geo-environment in which it
occurs. For example, if the geosphere is considered to be a geosphere with a fractal structure
[
35
]—very porous, heterogeneous media with complex pore space topology, then we can
arrive at anomalous radon-transport processes (Levy flights), such as superdiffusion [
36
].
These processes are investigated in the framework of the concept of spatial non-locality.
Non-locality in time leads to less intensive processes, for example, subdiffusion. In this
case, particles adhere to pore walls, which characterizes traps in which particles can stay
for a long time. Time non-locality is related to the notion of heredity (memory) [
37
,
38
]. It
means that particles, for some time, “remember” how they found their way into these traps.
Heredity effects occur in various systems, for example, in mechanical systems when
describing visco-elastic-plastic media [
39
], in biological systems - when describing the
viral propagation [
40
], in economic systems - when describing cycles and crises [
41
]. The
mathematical apparatus of heredity research is based on the theory of integrodifferential
equations developed by Italian mathematician Vito Volterra [
42
]. The peculiarity of inte-
grodifferential equations is that the difference kernel in the integrand has certain properties
(Volterra’s hereditary principles) and is called a memory function.
Let us consider radon accumulation in the chamber, taking into account heredity. Let
the dynamics of radon accumulation be described by the equation:
Zt
0K(t−τ)˙
¯
A(τ)dτ=F(¯
A(t),t), (5)

Geosciences 2025,15, 30 8 of 23
where
K(t−τ)
is a memory function;
F(¯
A(t),t)
is some function that is responsible for the
mechanisms of radon accumulation; ˙
¯
A(τ) = d¯
A(t)/dt.
Equation (5) is integrodifferential and describes the hereditary process of radon accu-
mulation in the chamber. If the memory function is chosen to be stepped, we can pass from
the integrodifferential equation to the equation with fractional derivative, the apparatus of
which is studied quite well [
43
,
44
]. The choice of the degree memory function is due to the
fact that degree laws are often found in nature [
45
], for example, such laws as Gutenberg-
Richter’s law, Amory’s law, etc. are known. In general, the memory function can be found
by solving the corresponding inverse problem from experimental data.
Let us choose a memory function in the following form:
K(t) = 1
Γ(1−α)θθ
tα
, (6)
where
Γ(
.
)
is the Euler gamma function;
θ
is some characteristic time of the process [
46
], [s];
0<α<1 is some parameter.
Substituting the ratio (6) into Equation (5), we obtain the following equation:
θα−1
Γ(1−α)Zt
0
˙
¯
A(τ)dτ
(t−τ)α=F(¯
A(t),t), (7)
Given the definition of the fractional Gerasimov–Caputo derivative of order 0
<α<
1,
which can be written as [47,48]:
∂α
0,t¯
A(t) = 1
Γ(1−α)Zt
0
˙
¯
A(τ)dτ
(t−τ)α. (8)
As a result, we obtain the following model equation:
θα−1∂α
0,t¯
A(t) = F(¯
A(t),t), (9)
If we put in Equation (9) that
F(¯
A(t),t)=λ0(1−¯
A(t))
, then we finally arrive at
the equation:
θα−1∂α
0,tA(φ) = λ0(1−¯
A(t)), (10)
For Equation (10), the local initial condition is true:
¯
A(t0) = ¯
A0. (11)
The problem (10), (11) admits an analytical solution in the form:
¯
A(t) = 1−(1−¯
A0)Eα−λ0θ1−αtα, (12)
where
Eα−λ0θ1−αtα=∑∞
k=0
(−λ0θα−1tα)k
Γ(1+αk)
—a special Mittag-Leffler function, whose prop-
erties are discussed in detail in [49].
Remark 4. Please note that when
α=
1, the Mittag-Leffler function becomes an exponential, i.e.,
E1(−λ0t)=e−λ0t
, and this leads us to the solution (4), which describes the accumulation of radon
in the chamber without taking heredity into account.
Remark 5. It should be noted that the order of the fractional derivative may depend on time,
which corresponds to the variable heritability [
50
,
51
]. In this case, the problem (10), (11) will be
generalized, and its solution should be sought by numerical methods [52,53].

Geosciences 2025,15, 30 9 of 23
In (Figure 3) in the Maple 2021 computer environment, the solution (12) was visualized
as a function of different values of
α
at different times
t
, [h]. It should be noted that the
calculated curves at small times at decreasing values of
α
indicate that the values of RVA
are sharply increasing. Furthermore, starting from some time instant, the calculated curves
regroup in the reverse order, forming steppe “tails”, which indicates a slower exit to
the asymptotics. The latter property is characteristic of nonlocal processes or processes
with heredity.
Figure 3. Calculated RVA curves plotted at
θ=
1 and different values of
α
: (a)
t∈[
0.5
]
; (b)
t∈[
0.500
]
.
Remark 6. Please note that in the case of spatial non-locality for the diffusion-convective model of
radon transport in the geosphere, the step tails in the calculated plots indicate an increase in the
diffusion length of radon. The fractional derivative on the spatial coordinate has here the order of
1
<α<
2, which indicates superdiffusion. The (Figure 4) shows an example of calculations by the
stationary diffusion-convective model of radon transport along the soil depth and their comparison
with the experimental data, which were investigated in the article [54].
In (Figure 4), it can be seen that the experimental curve is located closer to the curve
α=
1.6, compared to the curve for ordinary diffusion
α=
2, indicating the presence of
fractal properties of the soil.
Based on the above, it can be assumed that the order of fractional derivative
α
is
related to the permeability of the geo-environment. At the decrease of this parameter, the
permeability of the medium increases, and it is easier for radon to penetrate through the
ground into the accumulation chamber. Then, the mechanism of heredity is switched on,
which gives a slowing down of radon output on saturation. The latter is characteristic of
geo-mediated saturation.
In the present work, we use the above methodology to describe anomalous effects in
the RVA time series that precede strong seismic events in Kamchatka.

Geosciences 2025,15, 30 10 of 23
Figure 4. Radon concentration distribution curve by depth, drawn through the experimental points
of the work [
54
], and the family of calculated curves of radon concentration in fractal medium with
A∞=
2.5 [kBq/m
3
] as a function of the parameter
α
(a): experimental curve (1), 2 (2), 1.8 (3), 1.6
(4), 1.4 (5), 1.2 (6), 1 (7). Correlation field between the RVA values of the experimental curve and the
values of the calculated curves with α=2 and α=1.6 (b).
3. About Radon Monitoring
On the Kamchatka peninsula, a network of observation sites [
55
] has been deployed to
monitor
222Rn
and additional environmental parameters (temperature, pressure, moisture,
etc.) that can affect the resulting accumulation curve using accumulation chambers with
gas-discharge counters. When planning the locations of subsurface gas-monitoring network
stations in Kamchatka, special attention is paid to river valleys that run along crustal faults.
These zones are characterized by increased permeability, which creates favorable conditions
for subsurface gases to escape into the atmosphere, as described in the article [56].
Remark 7. Analysis of RVA data and related parameters obtained during continuous monitoring is
one of the methods of searching for earthquake precursors.
This is due to the fact that RVA is affected by changes in the stress-strain state of the
medium through which subsurface gas comes to the surface [
57
], so
222Rn
is considered a
well-known and well-proven indicator of the processes occurring in such a medium [
18
,
58
].
Monitoring of
222Rn
as a method of searching for precursors of seismic events has proved
itself in recent years [8], especially as a short-term precursor (up to 15 days) [59,60].
All points in the network are organized according to a common principle. The sensors
are usually located in accumulation chambers (10 L galvanized buckets) in the aeration
zone. When the equilibrium between
222Rn
and its decay products is reached in the
accumulation chamber, the intensity of
β
-radiation increases, which increases the sensitivity
of measurements. The insignificant convective component of subsurface air in the chamber
was ensured by holes in the bottom of the bucket and loose packing of the pipe through
which the detector was lowered into the chamber (Figure 5).

Geosciences 2025,15, 30 11 of 23
Figure 5. Scheme of placement of GDC-19 gas-discharge detectors for registration of
β
-radiation of
radon decay products.
Gas-discharge counters are the most common detectors of
β
- and
γ
-radiation. The high
sensitivity of counters allows the registration of single quanta of ionizing radiation, and
the large output signal is easily registered by recalculation circuits. All this allows passive
registration of
222Rn
in subsurface air by
β
-radiation of short-lived products of its decay
with a high degree of reliability and rather simple metrology. Conversion from pulses to
RVA is carried out by the empirical formula
A(t) =
9
·M
(Bq/m
3
), where
M
is the number
of pulses registered by
β
-radiation sensors per minute), obtained as a result of synchronous
registration by certified radiometers RS-410F by femto-TECH (USA), RRA-01M-03 (Russia)
and recorder with GDC-19 detector.
The data on (Figure 6) used in this work were obtained from the RVA sensor at the
INSR point installed in the aeration zone of loose sediments at a depth of 3 m from the
day surface [
7
]. On (Figure 6) shows the RVA and atmospheric pressure curves. On the
RVA curve for estimation of radon flux density, the two least noisy areas were selected for
exogenous impacts. In the second plot (Figure 6b), the abrupt release of RVA is associated
with the arrival of a cyclone and a decrease in atmospheric pressure. To use these data for
modeling purposes, the RVA data were removed at the specified site, and the skip was
filled in using the Lubushin A.A. software package [61].
The figure shows that three days before the earthquake, there was a sharp decrease in
RVA, and then followed by a new stage of accumulation. It is supposed that such behavior
of RVA is connected with the last stage of development of the source of future earthquakes
and is determined by the propagation of deformations in Earth’s crust. Similar behavior of
RVA curves was recorded before a number of strong earthquakes in southern Kamchatka
and earlier [7].
Data from the KRMR point were provided by Makarov E.O., senior researcher of
the laboratory of acoustic and radon monitoring of the Kamchatka branch of the Fed-
eral Research Center “Unified Geophysical Service of the Russian Academy of Sciences”
Petropavlovsk-Kamchatsky, Russia. The work was supported by the Ministry of Education
and Science of Russia (within the framework of the state task No. 075-00682-24) and with
the use of data obtained at the unique scientific installation “Seismoinfrasound complex for
monitoring of the Arctic cryolithozone and the complex of continuous seismic monitoring
of the Russian Federation, neighboring territories and the world”.

Geosciences 2025,15, 30 12 of 23
Figure 6. (green)—Experimental RVA data from the INSR observation site, in the temporal vicinity of
the 50-day earthquake: (a)—RVA
before earthquake
, during 27 July 2024 (16:00)—10 August 2024
(15:30); (b)—RVA after earthquake, during 16 August 2024 (16:00)—29 August 2024 (15:30); (grey)—
atmospheric pressure data at INSR point; (light blue)—excised section of the RVA due to the strong
influence of reduced pressure; (red)—7.0 magnitude earthquake.
4. Direct Problem for the Hereditary α-Model RVA
The (1) and (3) equations use ODE, which significantly limits the flexibility of the ODE
model. Therefore, the authors in [
27
,
62
] propose a modification of the (3) model consisting
of replacing the first-order ordinary derivative by a fractional derivative of [
43
,
44
] of
constant order.
Remark 8. There are other definitions of the fractional order derivative. They can be found, for
example, in scientific works [
50
,
51
,
63
]. However, the use of fractional Gerasimov–Caputo derivative
(8) is the simplest and, at the same time, quite effective approach.
A series of works by the authors of [
27
,
52
,
62
] is devoted to investigating issues related
to mathematical modeling of RVA, where it is assumed that the parameter
α
describes the
fractality of the [
35
] of the geo-medium and is related to its characteristics such as porosity,
permeability, and fracturing.
To numerically solve the problem (10), we will use the previously developed nonlocal
implicit finite-difference scheme (IFDS) [53] defined in a uniform grid domain:
h=T/N,b
Ω={(ti=ih): 0 ≤i<N},b
A∈b
Ω,
A(t) = Ai, 0 <Ai<1. (13)

Geosciences 2025,15, 30 13 of 23
Definition 1. Then, the difference direct problem:
Ai=1−d
∂α
0,ih Ai
λ0
,A0−const, 1 ≤i<N,A(0) = A0,
d
∂α
0,ih Ai=h−α
Γ(2−α)
i−1
∑
j=0(j+1)1−α−j1−αAi−j−Ai−j−1.
(14)
is a Cauchy problem consisting of finding a discrete function
Ai
in the region
b
Ω
with known
constants αand λ0.
Remark 9. IFDS scheme (14) and its generalized analogs have been tested in a number of tests and
applied problems [
53
,
64
]. Then (14) is solved by the modified Newton method (MNM), and then the
IFDS-based difference direct problem is unconditionally stable.
Remark 10. The hereditary
α
-model RVA (10) solved by the scheme (14) at the value
α=
1will pass
to the ODE model RVA (3), this is demonstrated in [
52
,
64
], which suggests that the generalization
of (10) is correct.
Next, in order to recover the values of the constant parameters
λ0
and
α
in the heredi-
tary
α
-model RVA, from the known experimental RVA accumulation data before (Figure 7)
and after (Figure 8) earthquake, the corresponding inverse problem [
65
,
66
] is formulated.
Figure 7. Experimental RVA data for 14 days, from the INSR observation point obtained before the
earthquake during the period: 27 July 2024 (16:00)—10 August 2024 (15:30).
Figure 8. (green)—experimental RVA data for 13 days, from INSR observation point obtained after
the earthquake (include earthquake) during the period: 16 August 2024 (16:00)—29 August 2024
(15:30); (red)—7.0 magnitude earthquake.
5. Inverse Problem on Parameters λ0and αfor the Hereditary
α-Model RVA
Previously, in [
27
,
52
,
62
] the parameters of models
λ0
and
α
were unknown and were
selected by brute force based on some considerations about the process flow, which is a

Geosciences 2025,15, 30 14 of 23
time-consuming approach. The accuracy of the selection was evaluated by the maximum
of
R2
—the coefficient of determination and
σ
—the Pearson correlation coefficient with
the maximum normalized experimental RVA data. This naturally leads to the search for
solutions to automate the selection of optimal parameters.
Let
Ai∈b
A
(and correspondingly
A(t)∈A
) be some function, but its solution depends
on the set of parameters
−→
X=[X0, . . . , XK−1]
, where
K=
2, and
X0=α
,
X1=λ0
. Let the
values of the discrete decision function
Ai∈b
A
are unknown, but additional information
(experimental data of RVA)
Ai=ϕi=−→
ϕ
on the solution of the direct Cauchy problem (14)
for the hereditary α-model of RVA is known.
Definition 2. Then the difference inverse problem for (14) is to restore the values
−→
X=[X0,X1]
from the known Ai=θiexperimental RVA data:
Ai=1−d
∂X0
0,ih Ai
X1
,Ai=θi, 1 ≤i<N,
d
∂X0
0,ih Ai=h−X0
Γ(2−X0)
i−1
∑
j=0(j+1)1−X0−j1−X0Ai−j−Ai−j−1.
(15)
To solve (15), we turn to the unconditional optimization theory [
67
]. For this purpose,
it is necessary to minimize the bias functional:
−→
η=−→
θ−ω(−→
X), minΨ−→
X=1
2
N−1
∑
i=0
η2
i=1
2
N−1
∑
i=0
(θi−ωi)2,(16)
where
−→
η
is a bias vector of dimension
N>K
, and the vector
ω(−→
X) = [ω0, . . . , ωN]
is a
vector of model data, i.e., the solution of the difference direct problem (14) with respect to
some approximation −→
Xobtained by solving the inverse problem.
The difference inverse problem is solved by a Newton-type unconditional optimization
method [68], namely the Levenberg–Marquardt iterative method [69], represented as:
∆X=−H−1×JT×−→
η,H=JT×J+γE, (17)
where
•∆X—the optimal increment of −→
Xfor the next iteration;
•Eis a unit matrix of dimension K×K;
•J=J−→
X
—a Jacobi matrix of dimension
N×K
with elements calculated by the
formula: Ji,k=∂ηi
∂Xk,i=0 . . . N−1, k=0 . . . K−1;
•
the derivative
∂ηi
∂Xk
is approximated by the difference operator
Ji,k=ηδ
i−ηi
δXk
, where
δX—a given small increment of −→
X;
•γ
is the regularization parameter of the method. If
γ∈R>0
and the Hesse matrix
H
is positive definite, then
∆X
is the direction of descent for the optimal step of the
method;
•
Start value:
γ(0)=v·max
idiagJX(0)T×JX(0)
, where
v
is a given starting
constant.
Remark 11. The solution of the inverse problem (15) by the Levenberg–Marquardt method (17),
hereafter (IP-LB), is reduced to starting from the given constants
X(0)
,
δX
,
v
, and
c
—constants for
recalculating
γ
during the loop, by recomputing the solution of the difference direct problem (14)
many times with approximations
−→
X
obtained during the solution of the inverse problem, to compute

Geosciences 2025,15, 30 15 of 23
the optimal values of
−→
X
. More details on the algorithm for implementing the optimization of the
vector −→
X can be found in the article [70].
The criterion for obtaining the optimal value is
ε≤Σ
, where
Σ
is the given accuracy
of IP-LB solution,
ε=1
N∑N−1
i=0η∆
i2
—mean square error (MSE) between experimental and
model RVA data.
All calculations related to the solution of direct and inverse problems by RVA models,
as well as calculations on data processing, were performed in the PRPHMM 1.0 software
package in MATLAB version R2023b for GNU/Linux Ubuntu Desktop 22.04. PRPHMM
1.0 is developed within the framework of the project “Modeling of dynamic processes
in geospheres taking into account heredity” at the expense of the grant of the Russian
Science Foundation
№
22-11-00064 (head Parovik R.I.) implemented at the Institute of
Cosmophysical Research and Radio Wave Propagation, Far East Branch of the Russian
Academy of Sciences, Paratunka village, Russia.
6. Radon Flux Density and Its Relation to Seismic Activity
As the experience of long-term observations has shown, before strong earthquakes
in southern Kamchatka, precursor anomalies are observed in the dynamics of RVA in the
subsurface air, the amplitude of which is no more than 30% of the background [
71
]. To
detect such precursor anomalies in the subsurface radon field in Kamchatka, as a rule,
in-phase variations are searched at several registration points, and the peculiarities of
the anomalies’ occurrence are investigated using mathematical modeling methods. For
example, in the paper [
72
] on the basis of a stationary mathematical model of diffusive-
advective radon transport it is shown that in an inhomogeneous layered medium with
increasing radon advection rate (
v
) the value of RVA increases proportionally, while the
radon flux density (RFD) increases according to a quadratic dependence. In a homogeneous
medium, when
v
increases by an order of magnitude, RFD also increases by an order of
magnitude, while RVA increases insignificantly. The theoretical estimates were confirmed
experimentally in [
73
], where it was shown that the dynamic characteristics of RVA in
soil air are less sensitive to changes in the stress-strain state of the geo-environment in
Kamchatka compared to RFD. As an example confirming this fact, we present the following
graphs (Figure 9).
Before the earthquake with magnitude
Mw=
6.2, which occurred on 24 August 2006
off the coast of southern Kamchatka at a distance of 190 km from the PRT point, there was
observed an anomaly characterized by the increase of RVA values on the ground surface
not more than 22% from the background (Figure 9a). At the same time, the calculated
values of RFD and
v
increase almost one and a half times (RFD by 51%, Figure 9b). The
radon flux density was calculated using the methodology proposed in the article [73].
It should be noted that using daily variations of atmospheric pressure as a probing
signal of the ground state, the permeability coefficient was calculated, which during this
period decreased almost 4 times: from 2.7
·
10
−13
to 6
·
10
−14
[m
2
] [
74
]. The increase of RFD
with a decreasing permeability coefficient testifies in favor of the fact that the mechanism
of mass transfer in the considered period cannot be described within the framework of the
classical diffusion-convection model. Therefore, it is possible that it is necessary to consider
loose sediments as a porous medium with fractal properties.
The paper [
75
] presents data on RVA and RDF variations at one of the network sites
before strong earthquakes in Kamchatka and concludes that the RFD parameter compared
to RVA variations was more sensitive to changes in the stress-strain state of the medium
before the deep Okhotomorsk earthquake with
Mw=
8.3. In addition, in the records of
radon flux density, a post-seismic effect associated with the change in ground permeability

Geosciences 2025,15, 30 16 of 23
after the 5-ball shaking caused by this earthquake in the vicinity of the observation point
was detected, which was not manifested in the RVA variations.
Figure 9. (a) RVA dynamics at the PRT point in the aeration zone at the depths of 0.1 and 1.0 [m] for
the period 20 July–29 August 2006; (b) calculated RFD and vadvection rate [74].
According to the [
7
] the arrival rate
222Rn
is described by the relation:
S=qΠ/V
.
Then, to estimate RFD based on a model stationary in the sense of the arrival rate
222Rn
(As
observed in Section 2.2), it suffices to express:
q=Amax λ0V
Π, (18)
where
•S=Amax λ0—the arrival rate of 222 Rn into the chamber, [Bq/m3s];
•q—RFD from the surface under the accumulation chamber, [Bq/m2s];
•Π—area of flow under the chamber (Figure 5), [m2];
•V—volume of the accumulation chamber (Figure 5), [m3].
Remark 12. The parameters of the accumulation chamber at the INSR observation point (Figure 5)
take the values: V =0.01,Π=0.05.
It is also important to note that when working with experimental data, RVA is normal-
ized to the maximum
Amax =
1 [Rel.unit]. However, if we want to estimate RFD for the
original RVA data, we should substitute the maximum un-normalized value in [Bq/m
3
]

Geosciences 2025,15, 30 17 of 23
in place of
Amax
. However, the values of AER
λ0
before the earthquake (Figure 7) and
after the earthquake (Figure 8) are unknown. However, by applying the method described
above, it is possible to recover the values of the parameters
α
and
λ0
in the model (10) on
the basis of experimental data on the variations of RVA in the accumulation chamber of
the INSR point, and then to estimate the values of RFD before the earthquake and after the
earthquake by (18).
Remark 13. Algorithms realizing solutions of direct problems: by ODE model (4) and hereditary
α
-model (10) underlying the IP-LB algorithm, make calculations in “meter/hour” values. This is
justified by the fact that the actual volume of the storage chamber (
∼
0.01 [m
3
]) and the frequency of
RVA registration (2 cycles/hour). However, for the convenience of perception, the other parameters
of the models (4) and (10) were recorded according to the international system of SI units. Therefore,
further on (Figures 10 and 11) the parameter values
T
,
t
,
h
in [h], and the recovered AER value
X1=λ0
in [h
−1
]. Hence, to estimate the RFD (18), we need to divide
λ0
by 3600 to go to the
scale [s−1].
Figure 10. Modeling results based on RVA data before the earthquake.
Figure 11. Modeling results based on RVA data after the earthquake.
Then for the camera in (Figure 5) RFD [Bq/m2s] will be calculated by the formula:
q=Amax λ0V
3600Π, (19)

Geosciences 2025,15, 30 18 of 23
7. Results of the Research
Visualization in this article was performed with the help of the FEVO 1.0 software
package, developed also in the Gnuplot 6.0 scripting language for GNU/Linux Ubuntu
Desktop 22.04. The FEVO 1.0 is developed within the framework of the project “Develop-
ment of a software package for modeling and analysis of volumetric activity of radon as
a precursor of strong earthquakes in Kamchatka” under the grant of the Russian Science
Foundation
№
23-71-01050 (supervised by D.A. Tverdyi) implemented at the Institute of
Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian
Academy of Sciences, Paratunka village, Russia.
Next, we will present the results of the IP-LB solution to recover the optimal values
of
X0=α
,
X1=λ0
of the stationary hereditary
α
RVA model (10) from the known
experimental RVA data (Figures 7and 8).
7.1. Results of Solving Inverse Problems
The results in (Figures 10 and 11) obtained with the following values of control
parameters:
hX(0)
0,δ0i=[0.05, 0.01]
for the index
α
; and
hX(0)
1,δ1i=[0.0025, 0.0005]
for
the coefficient
λ0
. The values of the control parameters
c
,
v
,
Σ
described above for the IP-LB
algorithm are shown in the figures.
The degree to which the model curves agree well with the experimental data was
evaluated by
σ
—Pearson correlation coefficient with the processed experimental RVA data
used for solving inverse problems. The key parameters characterizing the obtained results
are reduced to “meter/second” according to the international system of SI units and are
summarized in Table 1.
Table 1. Parameters of mathematical models: classical ODE (2), hereditary
α
-model RVA (10) solved
by methods of inverse problems, as well as similarity coefficients of model curves and data.
INSR
Dates Data
Sampling
Amax
[Bq/m3]
Set
λ0for
ODE
[s−1]
Restore
λ0for
Heredity
[s−1]
α
for
ODE
Restore α
for
Heredity
Correlation
for
ODE
Correlation
for
Heredity
RFD
by
ODE
[Bq/m2s]
RFD
by
Heredity
[Bq/m2s]
Before:
27 July 2024–
10 August 2024
2277.34 1.11 ·10−51.194 ·10−51 0.548 64 % 89 % 5.06 ·10−35.44 ·10−3
After:
16 August 2024–
29 August 2024
2400.17 1.11 ·10−51.277 ·10−51 0.573 60 % 78 % 5.33 ·10−36.13 ·10−3
Remark 14. The recovery of
λ0
is strongly influenced by the choice of its control parameters in
the IP-LB algorithm, i.e.,
X(0)
1
—the initial specified approximation and
δ1
—the initial specified
increment of
X(0)
1
. It was observed that if the initial approximation and its increment are set
±
1%
of the estimate of the maximum value of the parameter to be recovered, the efficiency and accuracy
of IP-LB will be the highest based on the estimates of the similarity coefficients for the described
data samples.
7.2. Sensitivity Analysis of the Solution
The first way to analyze the sensitivity of the solution to the inverse problem is to add
a small perturbation ϵ=10−4to the right-hand side of Equation (15) as follows:
Ai=1−d
∂X0
0,ih Ai
X1
,Ai=ϕi, 1 ≤i<N,A(0) = A0,
d
∂X0
0,ih Ai=h−X0
Γ(2−X0)
i−1
∑
j=0(j+1)1−X0−j1−X0Ai−j−Ai−j−1+ϵ.
(20)

Geosciences 2025,15, 30 19 of 23
The second way to analyze the sensitivity of the solution to the inverse problem is to
add a small perturbation ϵ=10−4to the initial condition (10) as follows:
Ai=1−d
∂X0
0,ih Ai
X1
,Ai=ϕi, 1 ≤i<N,A(0) = A0+ϵ,
d
∂X0
0,ih Ai=h−X0
Γ(2−X0)
i−1
∑
j=0(j+1)1−X0−j1−X0Ai−j−Ai−j−1.
(21)
The following conclusions can be made from the results of sensitivity analysis
(Table 2).
The algorithm for solving the inverse problem produces the first solution calculated by
it, which is closest to the optimal one. Furthermore, no matter how much we would not
reduce the threshold of accuracy
Σ
when fixing the other control parameters, the solution
algorithm “stuck”, giving the same solution. Moreover, by introducing a small perturbation
ϵ
to the inverse problem (20) or (21), the algorithm also “stuck”, but with some other
solution (with different values of
α
and
λ0
). This is very similar to the “local optimum trap”,
while the problem is to find the global optimum. Moreover, by introducing perturbations
ϵ
of different values (regardless of the order), the algorithm for solving the inverse problem
converges to different points of the local optimum.
Table 2. Sensitivity analysis of the formulated inverse problem (15).
The
Magnitude of
Disturbance
INSR
Dates Data
Sampling
Not
Disturbance
Solve Inverse
Problem (15)
With
Disturbance
Right
Hence (20)
With
Disturbance
Initial
Conditional (21)
10−427 July 2024–10 August 2024 1.194 ·10−50.548 0.97 ·10−50.526 0.88 ·10−50.556
10−416 August 2024–29 August 2024 1.277 ·10−50.573 0.94 ·10−50.577 1.30 ·10−50.563
10−527 July 2024–10 August 2024 1.194 ·10−50.548 1.08 ·10−50.57 1.66 ·10−50.525
10−516 August 2024–29 August 2024 1.277 ·10−50.573 1.44 ·10−50.592 1.19 ·10−50.545
10−627 July 2024–10 August 2024 1.194 ·10−50.548 1.58 ·10−50.55 1.36 ·10−50.539
10−616 August 2024–29 August 2024 1.277 ·10−50.573 1.36 ·10−50.544 1.0 ·10−50.608
The obtained analysis results do not necessarily indicate that the solution of the
posed inverse problem (15) is not the only one. However, it is clear that in the future it
is necessary to apply more specialized methods to recover the parameters of the model
equation. Nevertheless, a general trend in the recovered values can be seen, namely that
α
after the earthquake is larger than αbefore the earthquake when calculated with different
values of the
ϵ
perturbation. No such dependence can be established for the parameter
λ0
.
8. Conclusions
The scientific novelty of the presented paper consists of the fact that, for the first time,
a hereditary mathematical model has been applied to estimate the change in radon flux
density under changes in the stress-strain state of the medium under strong earthquake
conditions. Moreover, RFD estimates before and after a strong earthquake were given for
the first time.
The results presented in Table 1show that according to the proposed hereditary
α
-
model RVA, the radon flux density increased by 12.75%, while based on the ODE model
RVA, the radon flux density estimate shows an increase of 5.39%. The obtained estimates
agree with the existing RFD calculation theory.
The evaluations also showed that RFD increased after the earthquake, which is proba-
bly due to the impact of seismic waves on the ground in the area of the installation of the

Geosciences 2025,15, 30 20 of 23
storage chamber, as well as with the processes of relaxation of the geosphere arising after
the seismic event. This impact led to a change in the permeability of rocks in the area of
installation of the storage chamber and an increase in the total diffusive-convective flow
of geo-gas. Post-seismic anomalous changes in RVA associated with the impact of seismic
waves were recorded earlier in Kamchatka after the strong Zhupanovskoe earthquake (30
January 2016, Mw=7.2).
The continuation of the study can be connected with the application of the described
methodology of RFD estimation based on the results of parameter reconstruction of the
hereditary
α
-model RVA, but already for experimental data on other earthquakes. It is
also possible to complicate the model by adding a term to the equation to account for and
compensate for the effects associated with the influence of atmospheric pressure and its
variations on the RVA.
Also, the continuation of the research can be connected with the application of “neural
networks” to restore the parameters of the hereditary
α
-model RVA in order to avoid falling
into traps of local optimums (false attractors).
Author Contributions: Conceptualization, D.T., E.M., and R.P.; methodology, D.T. and E.M.; software,
D.T.; validation, D.T., E.M., and R.P.; formal analysis, E.M. and R.P.; investigation, D.T. and E.M.;
resources, D.T. and E.M.; data curation, E.M.; writing—original draft preparation, D.T. and E.M.;
writing—review and editing, E.M. and R.P.; visualization, D.T.; supervision, R.P.; project adminis-
tration, D.T.; funding acquisition, R.P. All authors have read and agreed to the published version of
the manuscript.
Funding: The study was carried out with the financial support of the Russian Science Foundation
(grants No. 22-11-00064 and No. 23-71-01050) and with the financial support of the Ministry of
Science and Higher Education of the Russian Federation (075-00682-24) and the financial support of
the State Assignment of IKIR FEB RAS (No. 124012300245-2).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The original contributions presented in this study are included in the
article. Further inquiries can be directed to the corresponding author.
Acknowledgments: The article is dedicated to the memory of Pavel Firstov, who made a significant
contribution to the study of volcanic and seismic activity in Kamchatka. Under the leadership of P.
Firstov, a radon monitoring network was organized in Kamchatka, and the theory of the emanation
method was developed. The authors are grateful to P. Firstov for the valuable ideas and advice he
gave when studying the topic of the article and are currently continuing his research.
Conflicts of Interest: The authors declare no conflicts of interest.
Abbreviations
The following abbreviations are used in this manuscript:
226Ra Radon
222Rn Radium
NEIC National Earthquake Information Center
UTC Coordinated Universal Time
SIS Seismic Intensity Scale
RVA Radon Volumetric Activity
GDC-19 Gas-Discharge Counter
RFD Radon Flux Density
AER Air Exchange Rate

Geosciences 2025,15, 30 21 of 23
ODE Ordinary Differential Equation
FD Fractional derivative
IFDS Implicit Finite-Difference Scheme
MNM Modified Newton’s Method
IP-LB Inverse Problem by method Levenberg–Marquardt
MSE Mean Squared Error
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