**4. Discussion**

#### *4.1. Justification of Reliability of FCAZ Recognition Results*

Simultaneously with the pure experiment (see above) or in the absence of the same, reliability was assessed based on the computational control experiments. FCAZ recognition employs two types of control experiments—individual seismic history and complete seismic history.

In the individual seismic history experiment, FCAZ zones are constructed based on findings from the DPS clustering of the earthquake epicenters (with *M* ≥ *M*R) only for 20 years preceding the events with *M* ≥ *M*0. The experiment ends with an analysis of the relative position of the recognized zones and the epicenter of the earthquake with *M* ≥ *M*0, for which the zones were constructed.

The complete seismic history experiment excludes from the used instrumental catalog the epicenters for the past few years during which events with *M* ≥ *M*0 have occurred. FCAZ zones are recognized through the use of DPS clustering of the epicenters remaining in the catalog. The experiment ends with an analysis of the location of the earthquake epicenters with *M* ≥ *M*0 from the discarded part of the catalog relative to the recognized FCAZ zones.

It should be noted that to improve the objectivity of computational experiments, they are conducted using the same values of the FCAZ method parameters (*q*, *β*, *δ*, *ω*, *v*, *C*) (i.e., the DPS and E2XT algorithms) as for the main recognition variant. The values *β*, *ω*, and *v* in the main recognition variant (see above) were computed in an automated manner by the artificial intelligence blocks [21].

Series of control experiments were conducted for the mountain belt of the South American Andes, the Pacific Coast of the Kamchatka Peninsula, California, and the Caucasus. Figure 11 shows typical results of experiments in California as an example.

**Figure 11.** California: (**a**) computational individual seismic history experiment for the earthquake dated 22 December 2003; (**b**) computational complete seismic history experiment (1960–1990) and the epicenters of earthquakes with *M* ≥ 6.5.

A comparative analysis of the spatial location of 29 FCAZ zones recognized in the individual seismic history experiments and the main results of FCAZ recognitions (see above) demonstrated a high degree of their similarity. That said, the epicenters of 27 out of 29 earthquakes involved in the experiments are located inside or at the boundaries of the recognized zones.

The FCAZ zones recognized in the course of complete seismic history experiments in terms of their forms and spatial location are close to the FCAZ zones of the main recognition variants. The zones include 25 out of 27 epicenters of earthquakes with *M* ≥ *M*0, which occurred years later (in particular, 10–25 years) after the date of the last recognition object (earthquake epicenter). For instance, in California, the epicenter of the earthquake with *M* = 7.1 (white star in Figure 11b), which occurred 28.5 years after the end of the catalog used in the experiment is located strictly inside FCAZ zones.

The results of control experiments demonstrate the stability of FCAZ recognition in time and space. This confirms the reliability of the main recognition variants in the studied regions as the zones prone to the strongest, strong, and significant earthquakes.

A comparative analysis of FCAZ zones and the EPA zones recognized earlier [4,62–64] was conducted in the mountain belt of the Andes, on the Pacific Coast of Kamchatka, in California, and the Caucasus. FCAZ zones typically occupy a smaller area than EPA zones. An exception is the mountain belt of the Andes, where FCAZ recognition covered a larger area. Figure 12 shows the comparison of FCAZ zones and EPA zones in Kamchatka and California.

In Kamchatka, high seismicity territories identified by both methods have a common, northeastern strike due to the subduction zone (Figure 12a). That said, FCAZ zones are typically located northwest of EPA zones. This is because most objects recognized as high seismicity ones using the EPA method were formed by the intersection of a deep-water trench with the morphostructural lineaments of rank II and III. At the same time, the main part of the epicenters that represent FCAZ recognition objects are located in the Benioff zone (seismic focal zone) within the continental slope before the trench and are generated by the convergen<sup>t</sup> interaction of two lithospheric plates.

The FCAZ zones are well aligned with the epicenters of known strongest (the Andes and Kamchatka), strong (California), and significant (the Caucasus) earthquakes. A check of this kind of alignment for EPA zones is not a clearly formulated objective. The reason is the construction peculiarities of morphostructural zoning scheme and the selection of EPA recognition objects, especially in the Pacific Seismic Rim regions. Moreover, EPA has no formalized transition from the classification of point objects to sought flat high seismicity zones with unambiguous boundaries. EPA solves this nontrivial problem by the trivial construction of circles with a radius proportionate to the magnitude of recognized earthquakes around the objects classified as high seismicity ones. Circles coincide with the areas initially used to compute the values of characteristics of the objects. The reasonableness of such transition is not obvious.

**Figure 12.** Comparison of the zones prone to earthquakes recognized by FCAZ and EPA methods: (**a**) Pacific Coast of the Kamchatka Peninsula; (**b**) California.

As regards the events with *M* ≥ *M0*, which constitute the material for pure experiment for both methods (FCAZ and EPA), seven out of eight epicenters of such earthquakes are located inside or at the boundaries of FCAZ zones. That said, only four epicenters are guaranteed to be located inside the EPA zones. It should be noted that the epicenter of the earthquake dated 6 July 2019, with *M* = 7.1 in California is situated strictly inside the FCAZ zones, ye<sup>t</sup> outside of the EPA zones. Summing it up, it is safe to say that the result of FCAZ recognition offers a whole range of benefits as compared with EPA results.

To ascertain the contribution of foreshock and aftershock sequences to the formation of the final result of FCAZ recognition, for the first time, epicenters from declustered catalogs were used as recognition objects. On the Pacific Coast of the Kamchatka Peninsula and in California, the FCAZ zones recognized based on complete and declustered catalogs turned out to be almost coinciding. This evidences that for the considered regions the existence of foreshock and aftershock sequences in the catalogs does not have a significant impact on the results of recognition of high seismicity areas as part of the FCAZ clustering method.

The optimal values of the parameter *β* computed automatically (the maximality of density in the DPS clusters, and in fact, the algorithm's "look" at the topology of the set of recognition objects and the separability of their dense condensations from the loose complement) for both recognitions in Kamchatka turned out to be very close: −0.2 and −0.2 for the declustered catalog; −0.15 and −0.2 for the complete catalog. Similar optimal values *β* in California are different. This can be explained by the fact that after a declustering of the catalog, the number of recognition objects went down by 68%, causing a change in the quantitative-spatial distribution of the set of objects. At the same time, the experiment in California can also be treated as successful since the results show that declustering a set of FCAZ recognition objects has not led to a significant change in either the DPS clusters or, in fact, the FCAZ zones [65].

#### *4.2. FCAZ Recognition as the Problem of Advanced Systems Analysis*

The FCAZ recognition problem is considered from the standpoint of advanced systems analysis [e.g., https://siiasa.ac.at/ access date: 30 July 2021]. The process and result of identification of potentially high seismicity hazard zones represent a complicated system [66]. The condition of the system depends on both spatial coordinates of recognition objects and on time. The results of FCAZ recognition obtained above follow from the algorithmic analysis of the currently identified objects *W* = {*w*}, which represent numerous epicenters of, generally speaking, fairly weak earthquakes.

For today, FCAZ performed a reliable recognition of sought high seismicity areas in several mountainous countries. Substantiations of such reliability are given for a certain period. This period is not long enough in both geological and real-time. In practice, it means tens, maximum hundreds of years. This period is characterized by the fact that a set of objects *w* ∈ *W* does not change drastically throughout the period. Here, a drastic change means not only the emergence of the clouds of new epicenters of earthquakes with *M* ≥ *M*R in previously aseismic areas but also significant alteration of the object distribution topology.

Let Δ*t* be a time interval during that the set *W* did not undergo any drastic changes. It is natural to assume that the FCAZ result obtained at the moment *t*1 will take place until the moment *t*2 = *t*1 + Δ*t*. Since *t*2, the set *W*, has significantly changed its spatial form and/or topology. Consequently, at the moment *t*2 it is necessary to perform a new FCAZ recognition taking into account the newly received initial data.

Treating this reasoning as the first step of the induction process also makes it easy to determine Δ*it* and the succession of the pairs:

$$\{(t\_i, \text{FCAZ}(t\_i)) : i = 1, 2, \dots\},\tag{18}$$

where time values *ti* are the moments when FCAZ recognitions are repeated.

It should be noted that generally speaking, Δ*it* = <sup>Δ</sup>*jt*, ∀ *i*, *j* = 1, 2, ... , *i* = *j* studying the dependence of Δ*it* on changes to the set *W* over time represents an independent nontrivial problem of systems analysis, which falls beyond the scope of this paper.

Accordingly, an analytical approach to the recognition of potentially high seismicity areas as a complex system that changes over time, even though stable over fairly long local intervals, was created in the present paper. The approach is based on the dynamic changes of the principal parameters of the system. The latter justifies the attribution of algorithmic succession *T*(*i*) × FCAZ, where *T* = {*ti*; *i* = 1, 2, . . .} is defined by the formula (18), to systems analysis methods. The general scheme of this method is illustrated in Figure 13, where *μi* is the measure of recognition quality at the moment *ti*.

**Figure 13.** Illustration of the systems analysis method developed based on FCAZ recognition. Deep red color shows the values of the FCAZ recognition quality measure at different time intervals.

Let FCAZ*γ*1 (*Wt*1 ) : *G* = *Bt*1 *Ht*1 be the result of FCAZ recognition ( *M* ≥ *M*0) at the moment *t*1. That said:


Let *<sup>B</sup>*0,*t*1 denote the set of epicenters of strong earthquakes that had occurred by the moment *t*1. It is obvious that the higher the value of the inclusion measure of epicenters *<sup>B</sup>*0,*t*1 in the subset of high seismicity objects *Bt*1, the better FCAZ recognition at the moment *t*1:

$$
\mu\left(B\_{0,t\_1} \subset B\_{t\_1}\right) = \left|B\_{0,t\_1} \cap B\_{t\_1}\right| / \left|B\_{0,t\_1}\right|.\tag{19}
$$

The quality of the FCAZ recognition problem considered above is determined by the fact that the results of future (after the appearance of new objects with time) expansions FCAZ*γ*(*Wt*) : *G* = *Bt Ht* tend to the limit characterized by the condition:

$$\lim\_{t \to \infty} \mu(B\_{0,t} \subset B\_t) \to 1. \tag{20}$$

Let us assume by the moment *t*2 = *t*1 + Δ*t Zt*1,*t*2 more strong earthquakes occurred, *<sup>B</sup>*0,*t*2 = *<sup>B</sup>*0,*t*1 ∪ *Zt*1,*t*2 , i.e., bringing their total number to *<sup>B</sup>*0,*t*2 = *<sup>B</sup>*0,*t*1 ∪ *Zt*1,*t*2 . That said, the total number of occurring earthquakes with M ≥ MR among the recognition objects increased by *zt*1,*t*2 to a total of *nt*2 = *nt*1 + *zt*1,*t*2 . Let us denote this new set of objects *Wt*2 = {*w*}. In this new situation, at the moment *t*1 + Δ*t* we have important additional information in place, which was not available to us at the moment *t*1. Accordingly, it is necessary to perform FCAZ recognition this time based on *Wt*2 to obtain the expansion FCAZ*γ*2 (*Wt*2 ) : *G* = *Bt*2 *Ht*2 (Figure 13).

FCAZ recognition is determined by the selection of free parameters *γ* = {*<sup>δ</sup>*, *C*, *ω*, *v*, *q*, *β*}. The parameters of the E2XT algorithm, as well as its result, directly depend on the recognized DPS clusters. In turn, *β* in DPS is the maximality level of density of DPS clusters, which depends on the spatial arrangemen<sup>t</sup> of objects. Due to the earthquakes with *M* ≥ *M*R, which occurred over the time Δ*t* = *t*2 − *t*1, the spatial distribution of objects *w* ∈ *Wt*2 will differ from the distribution of objects *w* ∈ *Wt*1 . For this reason, the selection of values *γ*2 = {*<sup>δ</sup>*, *C*, *ω*, *v*, *q*, *β*} for FCAZ*γ*2 (*Wt*2 ) : *G* = *Bt*2 *Ht*2 must be performed by the above-mentioned artificial intelligence blocks. These blocks ensure the selection of optimal values of input parameters accounting the spatial distribution of recognition objects at a given moment in time.

It is clear that the spatial distribution of a set of objects *Wt*2 can be so dramatically different from the spatial distribution of *Wt*1 that *Bt*1 will not be a proper subset *Bt*2 . In other words, the threshold (20) can fail to be achieved. To prevent this kind of situation and create a successive monotonous growing of FCAZ zones as the high seismicity areas recognized at the moment *t*2, the integration of the zones *Bt*1 and *Bt*2 should be taken, i.e., *Bt*2 = *Bt*2 ∪ *Bt*1 .

FCAZ recognition in the subsequent moments in time *tk* = *tk*−<sup>1</sup> + Δ*t*, *k* = 3, 4, ... is constructed similarly following the process of induction.

Based on the FCAZ results presented in this paper, the moment in time when control experiments were conducted (e.g., complete seismic history), for any of the studied regions is fixed as *t*1. Then *t*2 is the moment for which the main result of FCAZ recognition was obtained. Then the comparison of sets of recognition objects and FCAZ zones at the moments *t*1 and *t*2 allows concluding that in respect of all regions considered over the time intervals Δ*t* = *t*2 − *t*1, the sets of objects *w* ∈ *W* did not undergo any drastic change. Accordingly, given such fixed *t*1 and *t*2, the moment *t*2 is not ye<sup>t</sup> time for the performance

of new FCAZ recognition taking into account new initial data. In this situation, pure examination and computational control experiments gain special importance.

Similarly, the time has not ye<sup>t</sup> come for new FCAZ recognition either if we take *t*1 as the moments for which the main results of FCAZ recognition are obtained and take as *t*2, for instance, the year 2021.
