**2. Probabilistic methods for describing the seismic regime**

As noted above, in a seismically active region against the background of external disturbances, conditions are formed for the development of local nonlinear processes, which are described by methods of nonlinear dynamics. The final stage of such instability is its destruction, which is registered on the Earth's surface in the form of an earthquake. Therefore, in order to consistently fulfill an earthquake forecast and answer the questions "where and when" a structure will form in a special state, and "what energy will be released" when it is destroyed, you need to know the trajectory of the unstable system, which describes the evolution of the active structure of the source zone in the phase space with dimension equal to the number of variables describing the behavior of the source zone.

For such a description of earthquakes, it is necessary to create a model that includes the whole complex of phenomena of various nature, and it is also necessary to know the parameters of the macroscopic state of the structure, as well as the boundary conditions. But there is no such model. However, assuming that the structure of the source zone is an unstable nonlinear system, which is under the influence of external perturbations, then, according to the general principles of nonlinear dynamics, it can go into a special limiting state. In this case (purely theoretically), we must calculate various averaged characteristics on the actual trajectories of unstable systems, the dynamics of which has the nature of chaos [5].

The averaging functional is chosen as a certain probability. Consequently, initially nonlinear systems with chaos dynamics are described by probabilistic methods. Therefore, the transition of the system to one or another special state and its destruction itself is of a probabilistic nature. However, it is the destruction of unstable structures in the lithosphere in a seismically active region that is perceived as an earthquake, and this fact, therefore, also has a probabilistic character. Thus, by studying the result of the destruction of the nonlinear structure of the source zone as a random event, we ultimately study the seismic regime by probabilistic methods and,

ultimately, by indirect methods, study the dynamics of the structure of seismically active zones of the lithosphere.

The methods of seismological monitoring of a stress-strain geoenvironment related to the study of changes in the seismic regime can be extended if the method for calculating the probability distribution of earthquakes for various random events proposed in [6] and further developed in [7] is used to study it. This method is based on the axiomatic approach proposed by A.N. Kolmogorov in 1933 [8]. The application of this method to the catalog of Kamchatka earthquakes makes it possible to study the dynamics of the seismic regime using probabilistic methods for various regions both over the entire time period of instrumental observations and over various intervals lasting several years. With this approach, the catalog of seismic phenomena is represented as a probabilistic space of three objects. This allows us to consider each earthquake as a single outcome *ω<sup>i</sup>* in the space of elementary events Ω, the power of which is determined by the number of events *n* (catalog). In turn, each elementary event *ω<sup>i</sup>* in Ω is characterized by a system of random variables: energy class *Ki*, latitude *φi*, longitude *λi*, depth *hi*, and time *ti*. The time of a specific event in this model, as a random variable and having no mathematical expectation, is excluded from this system. In the future, we will consider a certain time interval Δ*T*, in which random events fall according to the catalog. The seismicity of the entire region or its selected part is considered as a complete group of events and is described in the form of distributions of conditional and unconditional probabilities *P* having a frequency representation. Random events are defined as combinations of a system of random variables *φ*, *λ, h,* and *K* in the set *F*~. This allows us to represent the catalog of seismic events over the observation period as a probability space of three objects {Ω*, F, P* ~ } and makes it possible to calculate probability distributions for various random events. If the distribution law of a system of random variables is given in analytical form by means of the distribution function *F*(*φ,λ,h,K*) or its density *f*(*φ,λ,h,K*)*,* then the distribution laws of individual variables can be found using standard formulas. In our formulation, the most logical is the reverse representation of the problem: using the laws of distribution of random variables, obtain the distribution law of the system. For continuous values of the probability of hitting random events for some interval, the time interval Δ*T* within the given intervals in latitude Δ*φi,* longitude Δ*λ*j, depth Δ*hm*, and class Δ*Kn* are calculated by the formula:

$$\begin{split} P\left(\Delta\rho\_{i},\,\Delta\lambda\_{j},\,\Delta h\_{m},\,\Delta K\_{n}\right) &= F\left(\rho\_{i},\,\lambda\_{j},h\_{m},K\_{n}\right) - F\left(\rho\_{i-1},\,\lambda\_{j-1},h\_{m-1},K\_{n-1}\right) = \\ &= P(\Delta\rho\_{i}) \cdot P\left(\Delta\lambda\_{j}|\Delta\rho\_{i}\right) \cdot P\left(\Delta h\_{m}|\Delta\rho\_{i},\,\Delta\lambda\_{j}\right) \cdot \\ &\quad P\left(\Delta k\_{n}|\Delta\rho\_{i},\,\Delta\lambda\_{j},\,\Delta h\_{m}\right), \end{split} \tag{1}$$

where *i, j, m*, and *n* are the indices of the corresponding intervals of random variables. This expression uses the notation: *P*(Δ*φi*) is the unconditional probability of events falling into the interval Δ*φi*; *P*(Δ*λj|*Δ*φi*) is the probability of occurrence of events in Δ*λ<sup>j</sup>* provided that the latitude of events is Δ*φi*; *P*(Δ*hm|*Δ*λj,*Δ*φi*) is the probability of hitting Δ*h*m, provided that the latitude and longitude are, respectively, Δ*φ<sup>i</sup>* and Δ*λj*; *P*(Δ*kn|*Δ*hm,*Δ*λj,*Δ*φi*) is the probability of falling into the interval of the energy class Δ*Kn,* provided that the longitude, latitude, and depth are Δ*λj,* Δ*φi*, and Δ*hm*, respectively. Numerical values of *P*(Δ*φi,* Δ*λj,* Δ*hm,* Δ*kn)* representations are easy to calculate. In a similar way, unconditional distribution laws are calculated for all random variables *φ, λ, h*, and *K,* as well as various combinations for conditional distribution laws from these variables. Processing the catalog according to the above

*Investigation of the Dynamics of the Seismic Regime in the Kamchatka… DOI: http://dx.doi.org/10.5772/intechopen.109069*

formula makes it possible to calculate the frequencies of seismic events in a given interval of variation of random variables Δ and obtain the values of the distribution function *F*(Δ*φi,*Δ*λj,*Δ*hm,*Δ*kn*)*.* Let us consider the practical application of this approach to describe the seismic regime using examples of the study of the distribution of the depth of weak events *KS* ≥ 8.5 on the eve of Kronotsky (1997-12-05).

To this end, on the basis of the described approach with the subsequent use of wavelet decomposition methods, we study the dynamics of changes in the probability distributions over the depth of "background" earthquakes that occur several years before strong Kamchatka events with *M* ≥ 7.0. This allows us to identify the depth range at which anomalous changes the parameters of this wavelet decomposition. At the same time, according to the findings of nonlinear dynamics, it is known that as the degree of instability of an arbitrary system increases and it approaches the critical state, both the intensity of parameter fluctuations and the time and length of correlations increase [1]. Therefore, the initial local ("microscopic") internal processes develop and acquire the character of coordinated ones, forming already on a global ("macroscopic") scale and capturing large seismically active areas. An increase in the length and amplitude of correlations in a nonequilibrium seismically active system indicates the connection of processes in some local selected area with its other parts. But logically, this should lead, over a certain time interval *τ*, to the formation of conditions conducive to an increase in the frequency of earthquakes in various parts of this region with less energy than in the main impending shock. Therefore, during the preparation of a strong (catastrophic) earthquake, large volumes of the lithosphere of a seismically active region are involved in the preparation area with a simultaneous increase in the frequency of occurrence of weak (background) events.

Based on the foregoing, it can be assumed that the preparation of an earthquake corresponds to the formation of an unstable nonlinear system, which is under the influence of external disturbance factors and develops according to the scenario of nonlinear dynamics. The interaction of the lithosphere with the environment (the chain "Sun—heliosphere—magnetosphere—ionosphere—neutral atmosphere"), with its nonequilibrium conditions, can be the starting point in the emergence of a new dynamic system, called the dissipative structure [9]. In this case, the scales of the connection between different parts of the nonlinear structure change. In other words, the scales of temporal and spatial correlation change. For example, during the formation of a dissipative structure, which are Benard cells or self-oscillating Belousov-Zhabotinsky reactions, the spatial scales change from intermolecular 10–<sup>8</sup> cm, which describe the interaction between molecules, to several cm [10, 11]. In turn, the time scales vary from 10–<sup>15</sup> s, corresponding, for example, to the periods of oscillations of individual molecules, to several seconds, minutes, or even hours [12]. With this in mind, we hypothesize:

During the preparation of the main major event with M 7.0 in a certain volume of a seismically active region, which is in an unstable state far from equilibrium, a consistent, correlated increase in seismic activity occurs at the "background" level, which covers areas far from the epicenter of the future event. At the same time, strong foreshocks with subsequent development of aftershock activity are possible in these areas at the depths of an upcoming earthquake. The scales that determine the temporal τ and spatial L correlation during the formation of these earthquakes are several years for τ and hundreds of kilometers for L and depend on the magnitude M of the upcoming main shock.

To test this hypothesis, the Kronotsky earthquake was considered: 1997-12-05 11:26:51 (UT), *φ* = 54.64° *N*, *λ* = 162.55° *E*, depth *h* = 10 km, depth determination error 2 km, energy class in terms of the amplitude of the *S* wave, determined by the nomogram of S. A. Fedotov *KS* = 15.5, local magnitude of the Kamchatka region (according to Ch. Richter) *ML* = 7.0, magnitude by code waves *Mc* = 7.7.

**Figure 1** shows the area along the eastern coast of the Kamchatka Peninsula, which is divided into 12 sectors, defined by intervals of latitude *Δφ* = 1° and longitude *Δλ* = 1.5°. For these areas, based on the catalog of seismic events provided by the KB GS RAS, on the basis of equation (1) the probability distributions *P*(Δ*h*) (histograms) characterizing the occurrence of seismic events with an energy class *KS* ≥ 8.5 (*M* ≥ 3.5) in the given depth intervals with a step *Δh* = 1 km will be further calculated up to *H* = 100 km for two cases—without taking into account the determination of the depth error and taking into account the given error. At the next processing step, the obtained seismic event probability distribution series *P*(Δ*h*) over depth were presented in the form of a continuous wavelet decomposition [4], which makes it possible to smooth the histogram of the probabilistic representation of the earthquake depth distribution:

$$P(\mathcal{W}\_{\Psi}P)(b,a) \coloneqq \left| a \right|^{-1/2} \int\_{-\infty}^{\infty} P(h) \overline{\Psi \left( \frac{h-b}{a} \right)} dt \tag{2}$$

where Ψ is the basis wavelet, *P*(*h*) is the numerical series of probabilities, and coefficients are *a*, *b*∈*R*, and *a*6¼0.

In the process of transformation, orthonormal Daubechies wavelets of the third order were used [4]. The decomposition was carried out up to the 32nd scale level. As an example, **Figure 2a** shows the original probability distribution series calculated for 1996 for the entire area indicated in **Figure 1**. **Figure 2b** shows the results of the wavelet transform of this distribution. Since the wavelet transform coefficients are proportional to the squares of the probabilities and, therefore, give the distribution of the intensity ("energy") of the process over scales [3, 4], the sum of the wavelet coefficients was calculated over all scale levels that characterize the distribution of the

#### **Figure 1.**

*Location of 12 regions along the eastern coast of Kamchatka with dimensions in latitude and longitude* Si *= Δ*φ�*Δ*λ *= 1°* � *1.5°.*

*Investigation of the Dynamics of the Seismic Regime in the Kamchatka… DOI: http://dx.doi.org/10.5772/intechopen.109069*

**Figure 2.**

*(a) Probability distribution earthquakes by depth for 1996 and (b) sum of wavelet coefficients for probability distribution earthquakes by depth for 1996.*

"energy" of the studied seismic process in depth in a given sector (y-axis in **Figure 2b**):

$$E = \sum\_{i=1}^{n} W\_{\forall} P\_i \tag{3}$$

where *n* is the number of scale decomposition levels and *W<sup>Ψ</sup> Pi* is the wavelet coefficient at the *i*th level of decomposition of the function *P*(*h*).
