Preamble

Large arthquake a Way of Formation and Prediction Zhiyong ZHU Shanghai Institute of Applied Physics, Chinese Academy of Science Jialuo Rd.2019, 201800, Shanghai, P.

China

Abstract

It is believed that the accumulation of small fractures (small earthquakes) in the crust is for the formation of large fractures (large earthquakes). In such cases, the temporal variations in the accumulation number of small earthquakes can be used to predict future seismic activity in the region.

To do so, a structural system of the crust is constructed using th e logarithmic linear relationship between earthquake frequency and magnitude, and a relationship between earthquake accumulation and time is derived by a ssuming that the rate of earthquake accumulation is proportional to the th power of the existing numb er of earthquakes . Earthquake record from selected regions of China and Italy are fitted by the theory and the obtained fitting parameters are used to evaluate the future seismic activity. It is found that the extent of deviation of earthquake accumulation from the theoretical expectation can be a reasonable judgement of the local seismic dangerous level ≥ 1 c an be considered a marker of local crust entering a ccelerated fracturing phase. proposed method physical evaluation of local seismic activity possible by simplifying the 3D (space magnitude) problem into 2D (time magnitude) problem in earthquake prediction

Keywords

Earthquake prediction, Seismic activity, Precursor, organized critic processes

1. Introduction

Earthquake could be severe disaster faced by humanity about 54% deaths caused all natural disasters . The exploration of earthquakes has a long history, modern scientific research beginning in the late 19th century. Although great progress has been achieved continuous improvements in detection technology and the development of theoretical analysis

methods

in the past 100 years, we remain still far from achieving the goal of earthq prediction Scientists expressed skepticism about the possibility of earthquake prediction from time to time and even come heated argument on this issue in the late 20th century The key points of the debate focus on two aspects. The first is whet her earthquake precursors exist?

How should earthquake precu rsors be defined and identified, what is their physical basis?

Zhiyong ZHU (E

The second is whether earthquakes are self organized critic al (or near critical) processe earthquakes inherently random? Discussions on the first type of question have generated many constructive insights, which standardize future screening and identification of earthquake precursors . However, there is no consensus on the second type of question. Some argue that the organized critical nature of earthquakes determines their randomness, suggesting that any small geological process could trigger a major earthquake . Others deny the randomness of earthquakes, asserting that the crust is not always in a self organized critical state, and that chaotic and nonlinear phenomena mainly occur during the unstable sliding phase . Furthermore, they argue that this self organized critical state can itself be considere d a precursor to earthquakes. While debates continue, the scientific community generally agrees that without a reliable fu ndamental theory of earthquakes it will be difficult to provide prediction services society ue to the diffic ulties in detecting logical structures deep underground , earthquake research has largely relied on a hypothesis verification approach, gradually approaching the truth through the process eliminating improper hypotheses.

The present work believes accumulation of small fractures (small earthquakes) over years in the crust is one of way for the formation of large fractures (large earthquakes).

It is also proposed that crustal structures, like other materials, are organized systems where smaller s tructures form larger structures, and larger structures form even larger ones. The so called logarithmic linear relationship between earthquake frequency and magnitude is merely an outward manifestation of the structural composition of materials unde r exte rnal forces. Based on these proposition , a theoretical relationship for the evolution of small earthquakes into large earthquakes has been derived, and to analyze earthquake data (here referring to the accumulation of earthquake occurrence numbers , the same below) from selected regions . The results show that the accumulation of small earthquakes in certain region can reflect the crustal energy level of local region , and the temporal change in small earthquake frequency can be used to jud future regional seismic activity.

2. Basic considerations

Earthquakes (here referring to tectonic earthquakes, the same below) are caused by fractures of crust resulting in rapid displacement between tectonic plates or crust blocks It is generally believed that crustal fractures occur when compressive forces between tectonic plates exceed their local fracture strength . This explanation accounts for earthquakes caused by compression and collision along plate boundaries. However, the source of fo rces driving earthquakes in inland regions far from plate boundaries remains uncertain.

Considering that the Earth is an approximately spherical structure with a solid crust enveloping a fluid like mantle he Earth's rotation and its periodic orbital motion around the Sun inevitably subject the crust to long term periodic impacts from the mantle.

These impacts can be direct generated by mantle dynamics, or indirect resulting from mantle induced movements. Such impacts are analogous to the forces e xerted on the inner walls of a rotating ment mixer by the cement inside. The that mountainous areas on the Earth's surface are more prone to earthquakes than plains support the proposition since the inner surface of the crust corresponding to mountainous area are structures protruding inward would more susceptible to mantle impacts compared to flatter regions.

Given the long term stability of Earth's structure and

orbital motion, it can be considered that this impact stress is stable an d with a fixed amplitude for each region. he periodic tidal fluctuations of the oceans can exert impact forces along ocean land boundaries and therefore may potentially producing similar effect Even if the stress intensity of such impacts is not high enough the continuous and repetitive nature of these forces over long periods can still cause damage to crustal structures, creating conditions for the downward migration of crustal material under the influence of gravity.

In other words, periodic impact s caused by the mantle result in the destruction of crustal structures. The destruction of these structures allows the crust to fracture and move under gravitational forces. The rapid relative movement between fractured crustal sections generates earthquak with magnitudes corresponding to the fracture . This process is similar to a high rise building collapsing after its foundational structure is gradually damaged. The collapse causes ground vibrations while enabling the transfer of material from reg ions of high gravitational potential to regions of low gravitational potential.

Of course, the apparent mechanical mechanisms of e arthquakes may vary among regions of the crust. For instance, as a tectonic plate or a crustal block undergoes fracturing and deformation due to mantle impacts, lateral pulling or compression forces may develop between it and adjacent plates or blocks. These continuously increasin lateral pulling or compression forces can also lead to fractures and earthquakes in boundary areas . However, since all these processes originate from mantle impacts that cause localized subsidence of material under the influence of gravity, the correspon intraplate earthquakes and plate /block boundary earthquakes are related and even follow the same Regarding the numerous self organized critical phenomena observed in nature, their essence likely lies in the inherent organizational structure o f matter. All matter consists of smaller structures forming larger structures, which in turn compose even larger structures in an organized hierarchy.

For this reason, when subjected to external forces, the macroscopic behavior of matter indirectly reveals its in ternal organizational framework, no matter whatever this process is critical (triggered and autonomously sustai ned) or non critical (one time), nothing but t he scale of the displayed organizational structure depends on the mode and intensity of the interactions involved.

For example, during a mountain blasting, the size distribution of fractured rock fragments corresponds to the original fine organizational structure of the mountain. However, the range of the displayed structure varies under differen t TNT equivalents . Similarly, repeatedly hammering a wall will lead to its collapse, and the number and size of the debris fragments will exhibit a clear logarithmic linear relationship. Yet, using hammers of different sizes or applying different levels of force will result in significant variations in the distribution range of the fragments. This can be understood as each structure having an optimal energy threshold for destruction. The hierarchical organization of matter determines that its absorption of external energy also occurs in a hierarchical manner (i.e. quantized manner) This should be similar to the quantization phenomenon we observe in the microscopic world. he Earth's crust absorbs energy mainly through two processes deformation and fracturing following the cyclic impact mantle continuously increasing static stress . Deformation corresponds to a slow energy absorption process, while fracturing corresponds to a rapid energy absorption with part of the absorbed energy released by causing earthquakes. released

energy by earthquakes should include the changes in local gravitation potential stimulated by the impact of mantle dynamics.

It is proposed , based on the consideration stated above, earthquakes of different magnitudes correspond to the fracturing of different structural levels within the crust. A magnitude 3 earthquake corresponds to the fracturing of a level 3 structure, a magnitude arthquake corresponds to the fracturing of a level 4 structure, and so on. Since a level 4 structure is composed of multiple level 3 structures, and a level 5 structure is composed of multiple level structures It means that withi n the same structural hierarchy the occurrence of multiple magnitude 3 earthquakes will lead to a magnitude 4 earthquake, and the occurrence of multiple magnitude 4 earthquakes will lead to a magnitude 5 earthquake, and so forth.

In other words, the occurrence of a large earthq uake in a region is the result of the accumulation of a sufficient number of smaller earthquakes, or rather, the destruction of a large structure in the crust is the consequence of the destruction of numerous smaller structures reaching a critical leve Clearly, a large earthquake originating in this manner does not require extremely high external forces.

This is similar to a building composed of many small structures; the gradual damage to these small structures over time eventually leads to the colla pse of the entire building. The taller the building , the greater the vibrations caused by its collapse. However, such a large scale collapse does not necessarily require significant external stress; the gravitational potential inherent in the building i tself is released as its smaller structu es are progressively destroyed.

3. Model construction

The occurrence probability of an earthquake in a specified region is a function of two independent variables: the earthquake magnitude and the occurrence time . If the magnitude probability density function is ) and the time probability density function is ), then the occurrence probability density function ) of an earthquake is:

The cumulative number of earthquakes with magnitudes greater than and less than (the maximum magnitude achievable in a region) that occur over the time interval from in the area can be expressed as:

It is known follows the Gutenberg Richter law the form (called seismic value) are constants.

Instead of constructing the probability density function of earthquakes, start constructing the crustal structure itself assuming that a level +1 crustal structure is composed of (here is the seismic value) level crustal structures.

Under the proposition that the destruction large structure is a result cumulative destruction of smaller structures, the number of magnitude +1 earthquakes generated by the des truction of level +1 structure should be 10 times the number of magnitude earthquakes produced by the destruction of lev structure That is Thus, the total number of earthquakes with magnitudes equal to greater than occurring in a region over a given period can be expressed as:

𝑛 𝑖 1 − 10 − 𝑏 ( for b> 0) ( 3 ) .

If the number of earthquakes of magnitude is defined as , where is the number of earthquakes of magnitude , then:

Because it is assumed here that earthquake magnitude is linked to the hierarchical structure of the crust, the relationship between the number of earthquakes and their magnitude reflects the spatial distribution of the local crustal structure.

Regarding the temporal characteristics of earthquakes, that is, the relationship between the number of earthquakes and time, it should, according to the perspective of this study, reflect the physical progressive process in which the successive destruction of small crustal structures leads to the failure of larger st ucture (i.e., the accumulation of small earthquakes eventually producing large earthquake Under external stress interaction , material typically exhibits two main processes of change. The first is a slow deformation process (a slow energy absorption pr ocess), which may sometimes gradually strengthen the material's structure by slowly altering it (e.g., the work hardening process in metals). The second is a rapid fracturing process (a fast energy absorption process), which weakens the material's structur e by causing its damage (e.g., the fracture process in materials). the Earth's crust, which is an intertwined structure of plastic and brittle materials, the mechanisms of structural strengthening and healing remain unclear. For now, we assume that the effect of external forces primarily results in the destruction of the crustal structure, disregarding any potential strengthening and/or healing effects that external forces might induce. This assumption is particularly reasonable during the later stages of accelerated crustal fracturing, as the slow processes of strengthening and healing in the crust would no longer be sufficient to counteract the fast fracturing process.

Since the destruction of small structures reduces the overall strength of a local cr , it makes the region more prone to structural fail ure under subsequent impacts of the same stress Therefore, it is reasonable to believe that the structural destruction of crustal regions under interaction periodic constant stress or continuou sly increasing static stress would be an accelerating process.

This implies that the frequency of small earthquakes will increase over time prior to the occurrence large earthquake.

To theoretically describe this increase in the frequency of small ea rthquakes, a simple approach is to assume that the rate of increase in the number of earthquakes of magnitude , denoted as is proportional to the cumulative number of earthquakes of the same magnitude that have already occurred. That is:

𝑑𝑛 𝑖 0 𝑑𝑡 = p × 𝑛 𝑖 0 (5 )

Here, represents the relative earthquake occurrence rate of earthquakes of magnitude in a region during the time interval expressed as = (dn . It is assumed here that s not change with time and is independent of magnitude. After integration of equation (5) , the number of earthquakes of magnitude occurring from time to time given by

𝑛 𝑖 0 = 𝑛 0 × 10 p ( 𝑡 − 𝑡 0 ) × 𝑙𝑔𝑒 (6 ) .

Where is the number of earthquakes of magnitude . Substituting equation (6) into equation (4), the total number of earthquakes of magnitude and above in the time period then given by

1 − 10 − 𝑏 = 10 𝑐 (8 ) ,

e have Formulas (7) and (9) reflect the relationship of the earthquake occurrence in a given region with the local crust geologic structure (earthquake magnitude) and the time.

Let the e xponential term in equation (9), , then:

This is the well known Gutenberg Richter law (G R relationship) . It is evident that function of time and is related to the relative earthquake occurrence rate and the value. In a given region, since , and are constants, will also have a fixed value for a given time interval. If the cumulative number of earthquakes of ma gnitude and above at time ), is set to 1, then from equation (9):

𝑡 𝑖 = 𝑡 0 +

This is the time corresponding to the first occurrence of a magnitude earthquake in the region.

If the maximum possible earthquake magnitude in the region is and the minimum earthquake magnitude is sufficiently small, then in equation (11) approaches the gestation or recurrence period the largest earthquake in the region , that

𝑡 𝑖 𝑚 − 𝑡 0 =

It should be noted that, due to the existence of an upper limit on earthquake magnitude in each region, value tends to increase in the high magnitude range. Therefore, when using equation (12) to calculate , the value should be corrected accordingly.

Using equation (9), the small earthquake data before the large earthquake in a region can be fitted to obtain the local seismic parameters, including , and . These parameters can then be used in equation (11) to calculate the future seismic trends in the region (i.e., to determine the time corresponding to the first occurrence of an earthquake of magnitude in the region).

During fitting, the constant e for the region (denoted as ) can be preliminarily determined using small earthquake data, and set . At the same time, the value can be set to zero (i.e., assuming that at , the cumulative number of earthquakes of magnitude = 0 and abov ), equals 1, see equations (8) and (9)). This leaves only as the two parameters to be determined during fitting.

It is important to note that when analyzing small earthquake data, the size and location of the selected area should be carefully considered.

Apart from not deviating too far from the core area, t area should neither be too small nor too large, If the area is too small or deviates from the core area the data of small earthquake may be incomplete, making it insufficie nt to reflect the entire preparation process of a single large earthquake, because the number of earthquakes of magnitude at any should satisfy formula (6) the area is too large, the data may come from tw o or more large earthquake preparation regions, thus exceeding the application scope of the theory Additionally, the time span of the selected data should be sufficiently long to include enough small earthquake data for fitting. It is also necessary to exclude aftershoc k data from previous major earthquakes in the area to avoid interference with the fitting process.

In reality, however, it is hard to know where the epicenter and how extensive scope of a large earthquake can be before a large earthquake occurred It is therefore a practical way to carry the analysis of earthquake data from area to area by assuming selected s are the independent large earthquake preparation zone and then make comprehensive analysis by synthesis of information got from different areas.

In this way the 3D (location magnitude) problem of earthquake prediction is simplified into 2D (time magnitude) problem which enables the evaluation seismic activities from area to area this certainly to locate the position of the preparation zone of a future large earthquake The above deriv ation is based on formula (5).

To make the theory more inclusive, we can assume that the number of earthquakes of magnitude increases with time at the power of of the cumulative number of earthquakes of that magnitude, i.e.:

𝑑𝑛 𝑖 0 𝑑𝑡 = p × 𝑛 𝑖 0 𝑞 (13 ).

So there is

𝑛 𝑖 0 = [ 𝑛 0 1 − 𝑞 + ( 1 − 𝑞 ) × p × ( 𝑡 − 𝑡 0 ) ]

, then we have the G R relationship fitting, can be fixed to 0 and to 1, so that the cumulative number of earthquakes of magnitude =0 and above at ), equals ). In this case, equation (15) is simplified

1 − 10 − 𝑏 × 10 − 𝑖𝑏 = 10 ( 𝑎 − 𝑖𝑏 ) (16 ) .

𝑎 = log {

Thus, the time corresponding to the first occurrence of an earthquake of magnitude in the region (i.e. )=1) is:

𝑡 𝑖 = 𝑡 0 +

The time period for the occurrence of the largest earthquake in the region ( magnitude ) is:

𝑇 𝑚 ≈ 𝑡 𝑖 𝑚 − 𝑡 0 =

For q>1, equation (19) shows that ll approximately equal at extremely large earthquake magnitudes Similarly, equations (15) and (16) reflect the relationship of the earthquake occurrence in a given region with the local crust geologic structure ( manifested by earthquake magnitude) and the time.

Using equation (16), small earthquake frequency data prior to a large earthquake in a region can be fitted to obtain the local seismic parameters, such as , and he constant value for the region (denoted as ) can be preliminarily determined u sing small earthquake frequency data) These parameters can then be used in equation (18) to determine the future seismic trends in the region (i.e., to estimate the time corresponding to the first occurrence of an earthquake of magnitude in the region).

Interestingly, with a fixed magnitude , equation (15) is very similar in form to the Paris model that describes the growth of micro cracks during the fatigue fracture process of metallic materials. The parameter here is equivalent to /2 in the Paris model.

The Paris model describes the steady growth process of cracks in metal materials under low cyclic stress (requires 2 < m < 4).

The continuous growth of these cracks eventually leads to the fracture of the material. Equating the earthquake magnitude with t he unit of crack length and the accumulation of small earthquakes with the growth process of crack length, a similar relationship can be derived.

4. Application

and discussion .1 Analysis of Earthquake Data in a Region of Sichuan Province, China ). Figure 2 [FIGURE:2] illustrates the longitude and latitude distribution of earthquakes in this region (longitude 104.4 106.0, latitude 27.5 30.0, marked as Zone I) during the 45 years. It is evident that a large number of earthquakes with magnitudes of 4 and above are distributed unevenly in time and space.

Because it is unclear how many major earthquakes are in preparation in this region, investigation was conducted not only the dat a from the entire region (Zone I) but also the data from several smaller subregions to examine the impact of area selection on the analysis.

These include the red boxed region (longitude 104.4 105.4, latitude 27.7 29.7, marked as Zone II) and the blue boxed region (longitude 104.4 105.4, latitude 27.9 28.9, marked as Zone III) in Figure 2, Based on the cumulative earthquake frequency data for magnitudes 4.0 and 4.5 in the entire region (Zone from 2000 to 2024, the value (defined as log( )/0.5) for this area is selected as 0.73±0.08 (see Figure 3 [FIGURE:3]). Therefore, in all subsequent fittings for data from various regions, the value is fixed at 0.73, and the impact of changes in the value on the results is analyzed as well.

China, from 1980 to 2024. China, from 1980 to 2024, and selection of analysis areas.

Estimation of the value for local earthquake data analysis I from 1980 to 2024 using equations (9) and (16), respectively, with the value being fixed at 0.73 during fitting. Since the earthquake data in Figure 4 [FIGURE:4] is based on records starting from 1980, the year 1979 is used as the starting point for data fitting. The cumulative earthquake quantity is expressed as the differential ∆ ) (see equati ons (20) and (21)), representing the total cumulative number of earthquakes from time minus the cumulative number of earthquakes expected before 1979.

When fitted using equation (9): When fitted using equation (16):

∆ 𝑁 𝑖 ( 𝑡 ) = { [ 1 + ( 1 − 𝑞 ) × p × ( 𝑡 − 𝑡 0 ) ]

(21). From Fig. 4, it can be observed that both equation (9) and equation (16) can fit the data. In particular, for the period between 2010 and 2024, the two models yield almost identical results. However, significant differences are observed in the results for the data prior to 2010 and the pr ted results beyond 2024. For the data prior to 2010, the fitting results from equation (16) alig n better with the actual data, whereas the predi ctions beyond 2024 remain to be evaluated by future observations. Table 1 [TABLE:1] shows the parameters obtained by fitting and compares the changes in fitting parameters for different values.

I. The solid line is the fitting result obtained using formula (9); the dotted line is the fitting result obtained using formula (16). 2024) in Zone I under different b values (with c fixed at 0).

Upper part with q=1 are fittings by using eq.(9). Lower part with q 1 are fittings by using eq.(16). value error error Figures 5a and 5b illustrate the relationship between the time of the first occurrence of an earthquake and magnitude , calculated using equations (11) and (18) based on the fitting parameters (table 1) obtained under different b values. It can be seen from the figures that the two models (equations (9) and (16)) produce similar predictions for the first occurrence time of earthquakes with magnitudes above 5, but they differ significantly in their predic tions for earthquakes with magnitudes below 5. This indicates that the two models exhibit notable differences in describing

the early stages of small earthquakes evolving into large earthquakes.

Additionally, as shown in Table 1 and Figure 5 [FIGURE:5], the ±0.08 var iation in the b value has little impact on the results under both models.

Zoomed in views (Figures 5c and 5d) reveal that, for different values, the predicted first occurrence times for a magnitude 7 earthquake in this region are approximately 2016 ( 65), 2019 ( =0.73), and 2023 ( =0.81), respectively.

The rule is that when M≥5, the smaller the value, the hi gher the magnitude of the earthquake occurred for the first time in the same period.

It should be noted that this calculation does not account for the effect of value increase among large magnitude earthquakes.

The adjustments to values for large magnitude earthquakes will be discussed in subsequent sections. value on earthquake trend predictions. ) Calculation results for Zone I data using equation (11). (b ) Calculation results for Zone I data using equation (18). (c) Magnified view of the red dashed box region in (a). (d) Magnified view of the red dashed box region in The left panel of Figure 6 [FIGURE:6] shows the fitting of M 4.5+ earthquake data from Zone I over three different periods starting 1980, 2000, and 2010 , respectively, to 2024 using equation (16) evaluate the impact of data collection p eriods on the fitting results.

Here, only the fitting results from equation (16) are presented, as the conclusions derived from fitting with equati on (9) are entirely consistent.

The right panel of Figure 6 presen ts the difference between the theoretical and observed values (i.e., the negative value of the residuals) to indicate the seismic hazard level of the region. is, when the theoretical value is significantly higher than the actual val ue, it means that t he earthquake accumulation in the local area is insufficient and there is potential for earthquakes to occur; when the theoretical value is significantly lower than the actual value, it means that the local crustal energy has been fully released and the curring possibility of another earthquake is reduced.

As shown in Figure 6, the fittings for the three time intervals yield similar residual values and patterns after 2010 indicating that meaningful results can be obtained even for relatively short datase ts (in this case, 15 years ) as long as the data is sufficiently abundant. As illustrated, the seismic hazard level in the region decreased significantly following a magnitude 5.7 earthquake in 2019, reaching its lowest point in 2020 before rapidly increasing again after phenomenon reflects the staged (i.e., quantized) energy absorption nature crust determined by the hierarchical structural

characteristics of the crust. As shown, the period from the deviation of the 2019 data from the theoretical value to the return of the 2023 data near the theoretical value lasted approximately five years. If the time required for the next jump is also estimated to be around five years ( i.e., assuming the theoretical value represents the statistical average of actual data), it suggests that a large earthquake might occur in the period around 2028. This means residuals can serve as a rough forecast signal of future earthquake trends, and such forecasts can be revised annually as more earthquake data becomes available. Table 2 [TABLE:2] presents the fitting parameters for M4.5+ earthquake data in Zone for the three different time intervals.

It can be observed that although the fitting parameters for different time intervals are similar, shorter time interval fitting yield larger parameter uncertainties I over different time intervals.

M4.5+ earthquake data in Zone I over different time intervals (with b=0.73 and c=0 fixed). time span error error The above discussion is based on the earthquake da ta for the entire region (Zone I) shown in the analysis region, it will be more helpful to determine the location of future earthquakes. Therefor e, the red framed region (Zone II) an d the blue framed re gion (Zone III) are selected from Zone I for data analysis (Figure 2) , where Zone I covers Zone II and Zone II covers Zone III. Table 3 [TABLE:3] compares the parameters obtained by fitting the data of earthquakes of magnitude 4.5 and above from 1980 to 2024 for the

three regions. The fittings were perform ed using equations (9) and (16), respectively. the corresponding relationship between the magnitude of the earthquake in the three regions and the time of its first occurrence calculated based on the parameters listed in Table 3. The left side shows the result calculated using formula (11) (Figure 7a [FIGURE:7]) and its local enlargement (Figure 7c), and the right side shows the result calculated using formula (18) (Figure 7b) and its local enlargement (Figure Overall, the first occurrence times for earthquakes with magnitudes of 6 or greater do not vary significantly across the three regions. Slight differences can be seen after local magnification. magnitudes below 8 (as the maximum earthquake magnitude in most crustal regions is typically below 8, calculations for M8+ are not discussed here), it can be observed that the larger the selected region, the earlier the first occurrence time for earthquakes of the same magnitude Specifically, the first occurrence times for M7 earthquakes, as determined from the data for Zone s I, II, and III, are 2019.2, 2020.4, and 2021.9, respectively, with differences from each other of 1.2 years and 1.5 years.

Figures 7a and 7c are derived from equation (11), which is based on equation (9) with assuming that there is only one major possibly the largest in the area earthquake preparation in the region. Howev er, if it is assumed that Zone I contains two major earthquake preparation points developing in sync (i.e., setting , with one located within Zone II and the other outside of it, then according to equation (9), the difference in the first occurrence times between the two regio would be approximately log(2)/0.4343/ ≈ 4.1 years (take =0.16876, see table 3) . Yet, as shown in rthquakes between Zone s I and II is only 1.2 years, a nd the difference between Zone s I and III is less than 3 years. This suggests that there is likely only one major earthquake preparation in the area. Alternatively, even if there is another paration , their development periods not overlap significantly, means that their mutual influence can be ignored. three regions (with b=0.73 and c=0 fixed).

Upper part are fittings with q=1 by using eq.(9). Lower part with 1 by using eq.(16). error error

Since the first occurrence times for M7 earthquakes determined from the three regions are very close, and the cumulative number of earthq uakes within Zone III accounts for more than 70% of the total in Zone I and more than 79% in Zone II, it can be inferred that the major earthquake preparation point for this cycle is likely located within Zone That is, if the data from Zone II were excluded, it would not be possible to deduce the occurrence of an M7+ earthquake in the near future for this region In fact, due to insufficient data remaining after excluding data in Zone , even the theoretical fitting described in th is study could not be performed (a) Results calculated by using parameters obtained with equation (9). (b) Results calculated by using parameters obtained with equation (16). (c) Magnified view of (a). (d) Magnified view of As mentioned above, the calculation results shown in Figures 5 and 7 do not consider the effect of the increased value occurred at large earthquakes. There is evidence that as the magnitude of earthquakes in each region approaches its limit, the number of earthquakes decreases significantly, which lead s to a rapid increase in the value. To reflect this phenomenon, the following relationship between the value and magnitude is proposed where and ∆ are variable parameters, and is the constant value for low magnitude earthquakes. The parameters used for correction in this paper are =0.01, =6.0 and ∆ =1.0. It should be noted that the correction here is only a demonstration, not a calcula tion based on facts. Formula ) and its parameters sh ould be optimized according to the actual situation in each place.

As is well known, the theoretical description of large earthquake behavior near the magnitude limit remains one of the central challenges in this field This paper only tak formula (22 ) as an example to illustrate how changes in the value will affect theoretical predictions. Figure 8 [FIGURE:8] shows the changes in the results of Figure 7 after the value is corrected as above.

It can be seen from the figure that the

increase in the value leads to a rapid increase in the first occurrence time of large magnitude earthquakes. As predicted by equation (18), with magnitudes high enough the first occurrence time of earthquakes will approach a constant value (Figures 8b and 8d).

Previousl y, based on the residual trend shown in Figure 6 , it was j udged that a large earthquake in the studied area might occur around

2028. If the

value correction here is reasonable, it indicates that the magnitude of the earthquake occurring around 2028 will be around 7.01 (see Figures 8c and 8d), while the corresponding magnitude of the earthquake without value correction will be around 7.76 (see Figures 7c and 7d). value correction). (a) Results calculated by using parameters obtained with equation (9). (b) Results calculated by using parameters obtained with equation (16). (c) Magnified view of Fig.8(a). (d) Magnified view of Fig.8(b).

Analysis of Earthquake Data in Italy year period (1980 2024) across a region covering almost the entirety of Italy (longitude 11 latitude 41 marked as Zone with data sourced from USGS boxed area in Figure 9a [FIGURE:9] (longitude 12.5 13.5, lati tude 42.0 43.5, marked as Zone II from 1980 t o 2024.

Figures 9 and 10, it is evident that a large number of M4+ earthquakes in the described region are unevenly distributed in both time and space. The reported values across Italy vary widely in the literature, which is related to differences in the geological structures of different areas.

In this paper, temporal variation of log(N was calculated using cumulative data for M4.0+ and M5.0+ earthquake in Zone I (see Figure 11 [FIGURE:11]). The average value from 2013 to 2024 was found to be 1.1 and is used as the fixed value for all subsequent fitting in various regions. Additionally, the impact taking =0.9 and =0.7 on the analysis was also investigated.

M4+ earthquakes in Italy and surrounding regions (marked as Zone I) from 1980 to 2024.

M4+ earthquakes in the red boxed region (marked as Zone II) of Figure 9a from 1980 to (Zone II) of Italy from 1980 to 2024.

M4.5+ earthquake data in Zone I from 1980 to 2024 using equations (9) and (16), respectively, being fixed at 1.1. Since the earthquake data in Figure 12 [FIGURE:12] is based on records starting from 1980, the year 1979 is used as the zero point for the earthquake data fitting. The cumulative earthquake quantity is expressed as the differential ∆ ) (see equations (20) and (21)), representing the total cumulative number of earthquakes f rom time minus the cumulative number of earthquakes expected before 1979.

From Figure 12, it is clear that both equation (9) and equation (16) can fit the data, and the value obtained from the fitting with equation (16) is very close to 1. Theref ore, only equation (9) is used for subsequent fittings.

I. The solid line is the fitting result obtained using formula (9), and the dotted line is the fitting result obtained using formula (16). 2024) in Zone under different values (with =0 fixed).

Upper part with q=1 are fittings by using eq.(9). Lower part with q 1 are fittings by using eq.(16). (year) error value error error

Figures 13a and 13b illustrate the relationship between the time of the first occurrence of an earthquake magnitude s I and II, respectively. The calculations are based on the fitting parameters obtained for M4.5+ earthquake data by using equations (11) and (18) under different values As can be seen from the figure , the changes the first occurrence of earthquakes caused value variation in the two zone very similar magnitudes 5, smaller values correspond to hi gher first occurrence magnitude the same period , and earlier first occurrence time for the same magnitude From the magnified views (Figures 13c and 13d), it can be observed that for Zone I, the predicted first occurrence times of M7 earthquakes are approximately 2055.6 ( =1.1), 2006.5 ( =0.9), and 1957.3 ( =0.7), respectively . For Zone II, the predicted first occurrence times of M7 earthquakes are approximately 2084.1 ( =1.1), 2037.9 ( .9), and 1991.6 ( =0.7), respectively. These results indicate that the large variation value has a significant impact on the prediction of future earthquake trends. It should be noted that these calculations do not account for the effect of value creasing associated with magnitude earthquakes. If this effect were considered, the corresponding first occurrence times of large earthquakes would be significantly delayed.

Results for Zone I data using equation (11). (b) Results for Zone II data using equation (11). (c) Magnified view of the red dashed box in (a). (d) Magnified view of the red dashed box in (b).

Additionally, as shown in Figure 13 [FIGURE:13], for the same b value, the first occurrence times of large earth quakes predicted for Zone II lag behind those for Zone I by approximately 30 year This is likely because Zone covers a larger area than Zone II and may contain 2 or more major earthquake preparation points during the same period, while the results in Figure 13 are calculated assuming that there is only one major earthquake preparation point within the If it is assumed that Zone II con tains only one major earthquake preparation point, while I contains two synchronized major earthquake prepa ration points (one inside Zone II and

another outside), then by using equation (9) with )=2, the first occurrence tim e of large earthquakes in Zone I would be delayed by approximately log(2)/0.4343/ =29.6 years (with =0.02341, see table 4 [TABLE:4]) compared to the calculation with )=1. This result matches the findings shown in Figure 13. presents the earthqu ake trend predictions for and II under =1.1 and =0.9, respectively, along with the corrected results assuming that Zone I contains two synchronized major earthquake preparation points during the same period. As shown in the figure, after co rrection, the results for Zone gn closely with those for Zone (see the magnified views in Figures 14c and 14d), supporting the hypothesis that there are likely two major earthquake preparation points in a large area of Italy during the same period, with ne of them located within As for whether there is only one major earthquake preparation point in Region II during the same period requires dividing Zone II into smaller subregions for data analysis.

Zone I (solid line) and Zone (dotted line). (a) Calculations based on parameters fitted with =1.1. (b) Calculations based on parameters fitted with =0.9. (c) Magnified view of the red boxed region in (a). (d) Magnified view of the red boxed region in (b).

The dashed line represents corrected results for Zone I (solid line) under the assumption that two synchronized large earthquake prepar ation points exist within Zone The left panel of s I and II from 1980 to 2024 using equation (20) (with =1.1 fixed). The right panel presents the difference between the theoretical fitting values and the observed values (i.e., the negative of the residuals) , which

are used to indicate the local seismic hazard level.

As can be seen from Figure 15 [FIGURE:15], the negative value of the residual well reflect s the historical events of large earthquakes in the two regions at various stages, indicating that a high negative value of the residual indicates an increased risk of a strong earthquake and can be used as precursor data. After 2017, the negative values of th residuals in the two s continued to increase, and by 2024, they had reached a level where an earthquake with a magnitude of 6 or greater could occur again, indicating that preparation earthquake disaster prevention and reduction to be made i n Zone I, both side and outside in recent years in Zone I and Zone II (with =0 fixed) error (right) of the lative earthquake data in Zone I (top) and Zone II (bottom).

The dangerous level of earthquake is defined as the negative value of the fitting residual. .3 Discussion

Stress interaction induces fracture of crust and therefore generating earthquakes. Large fracture s lead to large ear thquakes, and small fracture s lead to small earthquakes.

Although th accumulation of small fracture s is not a necessary condition for the formation of large fracture favorable c onditions for formation large fracture s and is certainly a important way to form large fracture Not only the continuously increasing static stress can produce small fractures, t he Earth's structure and its periodic motion create the condition for the crust to be periodically impacted by fluid mantle and therefore induce small fractures The accumulation of small fractures together with possible superposition of local gravity leads to large fracturing of the crust.

This makes it possible monitor formation large earthquake by observing the accumulation of small earthquakes. The fact that small fracture must accumulate to a certain level before a large fracture can form suggests that there is an effective interaction distance betwe fracture s; the accumulation of small earthquakes that are too far apart in favor of the formation of large earthquake . Areas where tectonic structure is conductive to stress concentration are more likely to generate closely spaced small earthquakes, and therefore are likely to become the preparation points for large earthqua mentioned previously, under stress interaction, the Earth's crust absorbs energy not only through crust fracturing. It can also absorb energy through crust deformation, a process of slow energy absorption.

If large fracturing (earthquake) of crust is mainly caused energy accumulation through this slow deformation pro cesses, then the proposed model will applicable because very few or even no small earthquakes can be detected b efore the large event occurred. process dominating the energy absorption of ea rth’s crust depend on the local geological structure of a region Taking the rate of increase in the number of earthquakes over time being proportional to th power of the cumulative number of earthquakes (equation 13) may be the simplest way describe the behavior of small fractures evolving into large fractures . The value obtained from fitting appears to serve as an indicator of different stages in earthquake evolution. When ≥ 1, it may indicate that earthquake preparation has matured and entered an accelerated growth phase, while < 1 may suggest that the large earthquake is still i n its early developmental stage, whereas 0 means that there are no t yet interactions (or correlation) between small fracture or that t local energy absorption process of crust dominated by crust deformation instead of fracturing q > 1, the relationship between the accumulation of earthquake numbers and time resembles the Paris model , which describes the growth of microcrack lengths during the fatigue fracture process of metallic materials. This similarity suggests that the formation mechanism of large earthquakes is analogous to the microscopic fracture mechanism during the final stage of fatigue fracture in metallic materia ls under low cyclic stress.

Metals, due to their excellent microscopic damage recovery mechanisms, only exhibit this damage acceleration phenomenon in the late stages of fatigue. It is expected that crustal materials, with their weaker damage recovery mech anisms, should enter this damage acceleration phase at a earlier stage.

The discontinuity (jumping nature) earthquake energy release of local crust provides a potential tool for earthquake prediction. The difference between theoretical predictions and

actual earthquake occurrences (negative residuals) can be used to assess seismic hazard levels in different regions. The accuracy of theoretical fitting improves as the amount of earthquake data increases annually, providing a natural basis for continues refining future seismic hazard assessments. The reliability of theoretical calculations depends on the quantity and quality of earthquake frequency data. ufficient, high ity earthquake data and a reasonable value as well as its correction for large magnitude earthquakes are critical obtaining reliable physics predictions.

5. Summary

Based on the Earth’s structure and its rotational and orbital dynamics, it is proposed that earthquakes may be a stepwise fracturing process of the crust under the impact of fluid mantle recurring annually together with the interaction of gravity Using th e logarithmic linear relationship between earthquake frequency and magnitude, a structural system of the crust is constructed, wherein smaller structures form larger ones and larger structures form even larger ones.

Since the accumulation of small fracture s (small earthquakes) in the crust is one of many ways for the formation of large fractures (large earthquakes) temporal variation model in the accumulation number of small earthquakes is derived by a ssuming that the rate of earthquake accumulation is proportional to the th power of the existing number of earthquakes. arthquake data from selected regions of China and Italy are fitted by the theory and the obtained fitting parameters are used to calc ulate the future trends in seismic activity in these region It is found that data analysis by the model from region to region simplified the 3D (space magnitude) problem into 2D (time magnitude) problem in earthquake prediction and can provide meaningful evaluation of local seismic activity. eviation of earthquake accumulation from the theoretical expectation can be used as a reasonable judgement of the local seismic dangerous level and ≥ 1 can be considered a marker of local t entering an accelerated fracturing phase.

It is believed that r eliable physics predictions can be conducted by the proposed model ufficient quality earthquake data and reasonable value as well as its correction for large magnitude earthquakes are available.

References

] Utsu T. list of deadly earthquakes in the world: 1500 2000., In: International handbook of earthquake and engineering seismology, Part A, San Diego: Academic Press, 2002,691 ] Engdahl E.R., Villasenor A., Global seism icity: 1900 1999, In: International handbook of earthquake and engineering seismology, Part A, San Diego: Academic Press, 2002,665 [3] Keilis Borok V. I., Soloviev A.A. (Eds.), Nonlinear dynamics of the lithosphere and earthquake prediction, Springer erlag Berlin Heidelberg 2003, ISSN 0172 7389, ISBN 978 ] Jordan T. H., Operational earthquake forecasting State of knowledge and guidelines for utilization, Annals of Geophysics , 54(4)(2011)316 384 and references therein

] Ullah Sh., Bi ndi D., Pilz M., Danciu L., Weatherill G., Zuccolo E., Ischuk A., Mikhailova N. N., Abdrakhmatov K., Parolai S., Probabilistic seismic hazard assessment for central Asia, Annals of Geophysics , 58(1)(2015)S0103; doi: 10.4401/ag ] Main I., Statistical Physics, Seismogensis, and Seismic Hazard, Reviews of Geophysics 34(4)(1996)433 462 and references therein. ] Geller R.J., Jackson D.D., Kagan Y.Y., Mulargia F.

Earthquakes cannot be predicted, Science 275(14)(1997)1616 references therein Wyss M., Cannot earthquakes be predicted?

Science )(1997) and references therein Aceves R. L., Park S. K., Science )(1997) and references therein Geller R.J., Jackson D.D., Kagan Y.Y., Mulargia F., Science )(1997) references therein ] Kangan Y.Y., Are earthquakes predictable?

Geophys. J. Int. 131(1997)505 references therein ] Sykes L.R., Shaw B.E., Scholz C.H., Rethinking earthquake prediction, Pageoph. 155(2) (1999) [13] Reid H. F., The mechanics of the earthquake, The California earthquake of April 18, 1906, Report of the State Investigation Commission, Vol.2, Carnegie Institute of Washington, Washington, D. C. 1910 ] Kerr R. A., Earth’s inner core is running a tad faster than the rest of the planet, Science , 309(5739) (2005) ] Gutenberg, Richter, Magnitude and energy of earthquake, Ann Geofis , 1956, 9:1 Republished in:

Annals of Geophysics , 53(1)(2010)7 12, doi:10.4401/ag ] Paris, P.C., Gomez M.P., Anderson W.P., 1961, "A rational analytic theory of fatigue," Trend in Engineering Vol. 13, pp. 9 search Isabel Serra & Álvaro Corral, Deviation from power law of the global seismic moment distribution, Scientific Reports , 2017 | 7:40045 | DOI: 10.1038/srep40045

A Mode of Nucleation and Its Physical Prediction

Shanghai Institute of Applied Physics, Chinese Academy of Sciences

The study of nucleation processes is fundamental to understanding phase transitions in condensed matter physics. In this paper, we explore a specific mode of nucleation and provide a theoretical framework for its physical prediction. By analyzing the thermodynamic stability and kinetic pathways of the system, we aim to characterize the critical parameters that govern the transition from a metastable state to a stable phase.

[FIGURE:1]

The nucleation rate is traditionally described by Classical Nucleation Theory (CNT), which posits that the formation of a new phase is driven by the competition between bulk energy gain and surface energy cost. For a spherical nucleus of radius $R$, the Gibbs free energy change $\Delta G$ is given by:

$$\Delta G = 4\pi R^2 \gamma - \frac{4}{3}\pi R^3 \Delta g_v$$

where $\gamma$ represents the surface tension and $\Delta g_v$ is the difference in free energy per unit volume between the two phases. The critical radius $R^$ and the activation barrier $\Delta G^$ are determined by the condition $\frac{d\Delta G}{dR} = 0$. However, in complex systems or under extreme conditions, the classical model often requires significant corrections to account for local fluctuations and non-equilibrium effects.

[TABLE:1]

In our proposed model, we consider the influence of external fields and structural constraints on the nucleation pathway. We hypothesize that the nucleation process follows a specific precursor-mediated mechanism, where a dense liquid-like cluster forms before the final crystalline structure emerges. This multi-step process can be represented by the following kinetic equations:

$$\begin{aligned} A &\xrightarrow{k_1} B \ B &\xrightarrow{k_2} C \end{aligned}$$

where $A$ is the initial metastable state, $B$ is the intermediate precursor, and $C$ is the stable phase. The transition rates $k_1$ and $k_2$ are sensitive to the temperature $T$ and the chemical potential $\mu$. By employing machine learning algorithms to analyze molecular dynamics trajectories, we can identify the local order parameters that distinguish these states with high precision.

[FIGURE:2]

Furthermore, the physical prediction of the nucleation time $\tau$ involves calculating the mean first-passage time (MFPT) across the energy landscape. Using the Fokker-Planck equation, we

摘要

Introduction

A prevailing view in seismology suggests that small-scale crustal ruptures (small earthquakes) can evolve into large-scale ruptures (large earthquakes). Under this premise, the temporal variation in the frequency of small earthquakes serves as a critical indicator for assessing future seismic activity within a specific region. This study utilizes the log-linear relationship between earthquake frequency and magnitude (the Gutenberg-Richter law) to analyze seismic trends. Given that the cumulative rate of seismic activity is proportional to the overall earthquake frequency, we investigate these patterns within Sichuan Province, China.

Methodology and Data Analysis

The fundamental framework of this research rests on the statistical properties of seismic sequences. Specifically, we examine the $b$-value and the cumulative seismic rate to characterize the stress state of the crust. According to the Gutenberg-Richter relationship:

$$\log_{10} N = a - bM$$

where $N$ represents the number of earthquakes with a magnitude greater than or equal to $M$, and $a$ and $b$ are constants. The $b$-value, in particular, is often inversely correlated with the differential stress in the crust. By monitoring fluctuations in the frequency of small-magnitude events over time, we can identify periods of seismic quiescence or acceleration that may precede significant tectonic shifts.

[FIGURE:1]

In the context of Sichuan Province, a region characterized by complex fault systems such as the Longmenshan fault zone, the relationship between cumulative seismic energy release and time is of paramount importance. We assume that the seismic cumulative rate is proportional to the underlying tectonic loading rate.

Regional Context: Sichuan Province

Sichuan Province is situated at the eastern margin of the Tibetan Plateau, making it one of the most seismically active regions in China. The interaction between the Qinghai-Tibet block and the Sichuan Basin creates intense crustal deformation. Our analysis focuses on the historical seismic data from this region, applying the aforementioned log-linear models to evaluate whether the current frequency of small earthquakes indicates an impending large-scale rupture.

[TABLE:1]

The data suggests that variations in the $a$ and $b$ parameters are not merely stochastic but are linked to the preparation phases of major seismic events. By integrating the cumulative rate of small earthquakes, we aim to provide a more robust predictive framework for seismic hazard assessment in the region.

Conclusion

By analyzing the temporal evolution of small earthquake frequency and its adherence to the log-linear magnitude-frequency relationship,

分析

...obtained local seismic fitting parameters and subsequently analyzed the development trends of future earthquakes. The difference between theoretical predictions and the actual number of seismic events—defined as the seismic hazard value—can serve as a reflection of crustal stability. This method of fitting regional seismic data contributes to...

1. 引言

Earthquakes represent a major natural disaster that humanity has long confronted \cite{1, 2}. The history of human inquiry into earthquakes is extensive, with modern scientific research commencing at the end of the 19th century. Although continuous improvements in detection technology and the development of theoretical analytical methods have progressively deepened our understanding of seismic phenomena, we remain far from achieving the goal of earthquake prediction today, more than a century later \cite{5, 6}. At one point, scientists even began to doubt whether earthquakes could be predicted at all, leading to an intense debate on the subject at the end of the twentieth century.

The core of this debate focuses on two primary issues. First, do earthquake precursors actually exist? Furthermore, how should these precursors be defined and identified, and what is their underlying physical basis? Second, is the occurrence of an earthquake a self-organized critical (or near-critical) process?

Is the occurrence of an earthquake a random event? Discussions regarding the first issue have yielded many constructive insights, establishing standards for the screening and identification of future earthquake precursors. Regarding the second issue, no consensus has yet been reached. Some researchers argue that the self-organized criticality of earthquakes determines their randomness, suggesting that any minor geological process has the potential to trigger a major seismic event \cite{7, 11}. Conversely, others deny the inherent randomness of earthquakes, asserting that the Earth's crust is not always in a state of self-organized criticality. They argue that chaos and nonlinear phenomena primarily appear during unstable sliding phases and that such self-organized critical states can themselves be regarded as earthquake precursors. Although the debate over the predictability of earthquakes continues to this day, the scientific community agrees that without the establishment of a reliable fundamental scientific theory of seismology, it will be difficult for humanity to provide effective technical services for earthquake prediction.

Most seismic research that yields data available for analysis adopts a method starting from...

方法

Abstract

Based on the internal structure of the Earth and the laws governing its rotation and revolution, this work proposes that earthquakes result from the year-by-year periodic and progressive fracturing of the crustal region caused by internal fluids and/or their secondary processes. Through this fracturing process, matter transitions from high-energy states to lower ones. Given that the crustal structure, like all matter, is an organizational system where smaller structures compose larger ones, and larger structures compose even greater ones, the well-known log-linear relationship between earthquake frequency and magnitude is merely an external manifestation of this structural composition under the influence of external forces.

On this basis, we derive a theoretical relationship describing the evolution of small earthquakes into large earthquakes and utilize this theory to analyze the cumulative frequency of earthquakes over time. The results demonstrate that the degree of accumulation in the number of small earthquakes within a region reflects the evolutionary state of its internal structure. Furthermore, temporal changes in the frequency of small earthquakes can be used to predict the developmental trends of future seismic activity in that region.

2. 基本思路

Earthquakes are caused by the rapid displacement resulting from the rupture of the Earth's crust. It is generally believed that crustal rupture occurs when the stress between tectonic plates exceeds the local fracture strength. While this explains phenomena at plate boundaries, the source of stress for intraplate earthquakes far from these boundaries remains a subject of debate. Considering that the Earth is a near-spherical structure consisting of a solid crust enveloping a fluid mantle, the crustal structure is governed by the dual effects of gravity and the outward support provided by mantle material. The Earth's periodic rotation and orbital motion around the sun inevitably subject the crust to long-term periodic stresses caused by these motions. These stresses can be generated directly by the mantle or through indirect effects of mantle movement. This process is analogous to the impact of cement against the inner walls of a rotating drum on a cement mixer. Given that the concave-convex structure of the crust's inner surface mirrors the Earth's outer surface, protruding internal structures are more susceptible to the impact of mantle material than flatter areas. This provides a potential explanation for why mountainous regions on the Earth's surface are more prone to earthquakes than plains.

Similar phenomena occur regarding the impact on land-sea boundaries. Even if the intensity of this stress is not high, its long-term, cyclical application can damage the crustal structure, creating conditions for the downward movement of crustal material under the influence of the gravity field. That is to say, when the equilibrium maintained by the mantle is disrupted, the resulting structural failure leads to crustal movement. The rapid relative displacement during this rupture generates an earthquake, much like a building collapsing after its structural integrity is compromised. This vibration occurs as the system transitions from a state of high gravitational potential energy. Simultaneously, as the crust deforms, lateral pulling forces are formed between adjacent sections. These lateral forces can also lead to rupture and earthquakes within plate interiors. These processes all originate from the local redistribution of mantle and crustal material under the influence of gravity.

Intraplate earthquakes and plate-boundary earthquakes even follow similar laws. Regarding the numerous self-organized criticality phenomena observed in nature, their essence should reflect the fact that the organizational structure of matter is inherently hierarchical; that is, all matter is composed of multiple small structures forming a larger structure, which in turn combine to form even larger organizational entities. For this reason, when matter is subjected to external forces (perturbations), its macroscopic behavior indirectly reveals its own organizational architecture. Whether this process is critical (self-sustaining after triggering) or non-critical (a one-time event), the revealed organizational architecture depends on the mode and intensity of the interaction. For example, during mountain blasting, the size distribution of fragmented rock debris aligns with the original organizational structure of the mountain, though the range of structures revealed varies under different explosive yields. Similarly, repeatedly hitting a wall with a hammer will eventually cause it to collapse; the relationship between the number and size of the resulting fragments will show a clear log-linear relationship, but the distribution range of these fragments will differ significantly depending on the size of the hammer and the force applied. This can be understood as there being a specific energy threshold required for the destruction of each structural level. The organizational architecture determines that the absorption of external energy is hierarchical, or "quantized," which is similar to the phenomena we observe in the microscopic world.

Based on the aforementioned reasoning, this paper proposes that after being subjected to impulsive effects caused by the mantle, the crust absorbs energy through two processes: deformation and rupture. Deformation corresponds to a slow energy absorption process, while rupture corresponds to a rapid energy absorption process accompanied by the occurrence of an earthquake.

The occurrence of earthquakes of different magnitudes corresponds to the rupture processes of different structural levels within the crust. Specifically, a magnitude $M$ earthquake corresponds to the rupture of a level-$M$ structure in the crust, while a magnitude $M-1$ earthquake corresponds to the rupture of a level-$(M-1)$ structure, and so on. Since a level-$M$ structure is composed of multiple level-$(M-1)$ structures, and a level-$(M-1)$ structure is composed of multiple level-$(M-2)$ structures, at the same structural level, the occurrence of multiple magnitude $M-1$ earthquakes will eventually lead to a single magnitude $M$ earthquake.

The occurrence of multiple magnitude $M-2$ earthquakes leads to a magnitude $M-1$ earthquake, and so forth. In other words, the occurrence of a large earthquake in a region is the result of the accumulation of a certain number of local small earthquakes. Alternatively, the destruction of a large regional structure is the result of countless small local structures being damaged to a certain extent. Obviously, large earthquakes generated in this manner do not require extremely high external forces. This is analogous to a building composed of many small structures; the cumulative damage to these small structures over time leads to the eventual collapse of the building. The vibration caused by the collapse allows the inherent gravitational potential energy to be released as the structure is progressively destroyed. The occurrence of an earthquake is a manifestation of this change in state. Because this process is an inherent property of the system, it remains difficult to measure precisely.

3. 模型构建

Probability Density of Earthquake Occurrences

In the study of seismic risk and hazard analysis, the occurrence of an earthquake is typically modeled as a function of independent variables, most notably the time of occurrence and the magnitude (intensity) of the event. If we assume that the timing of an earthquake and its magnitude are independent random variables, we can define their respective probability distributions to derive the joint probability density of a seismic event.

1. Fundamental Probability Density Functions

To characterize the nature of seismic activity, we define two primary probability density functions (PDFs):

• Temporal Probability Density Function: Let $T$ be the random variable representing the time of earthquake occurrence. The probability density function of the occurrence time is denoted as $f_T(t)$. In many seismological models, such as the Poisson process, the inter-arrival times between events are often modeled using an exponential distribution, whereas specific time-dependent models might utilize Weibull or Lognormal distributions.

• Magnitude Probability Density Function: Let $M$ be the random variable representing the magnitude of the earthquake. The probability density function of the magnitude is denoted as $f_M(m)$. This is commonly derived from the Gutenberg-Richter law, which describes the relationship between the magnitude and the total number of earthquakes in any given region and time period.

2. Joint Probability Density of Earthquake Events

Under the assumption that the time of occurrence $T$ and the magnitude $M$ are independent variables, the joint probability density function $f_{T,M}(t, m)$ for an earthquake occurring at time $t$ with magnitude $m$ is the product of their individual marginal densities:

$$f_{T,M}(t, m) = f_T(t) \cdot f_M(m)$$

This independence implies that the energy released (magnitude) does not depend on when the event occurs, and conversely, the timing of the event is not influenced by the potential magnitude of the earthquake.

3. Probability of Occurrence within a Specific Interval

The probability $P$ that an earthquake will occur within a specific time interval $[t_1, t_2]$ and fall within a specific magnitude range $[m_1, m_2]$ can be calculated by integrating the joint probability density function over the defined domain:

$$P(t_1 \le T \le t_2, m_1 \le M \le m_2) = \int_{t_1}^{t_2} \int_{m_1}^{m_2} f_{T,M}(t, m) \, dm \, dt$$

𝜌 ( 𝑖 , 𝑡 ) = 𝑓 ( 𝑖 ) × 𝑓 ( 𝑡 ) (1).

By applying the probability density function of Richter's Law, we can construct a model of crustal structure wherein the crust of a specific region is composed of $n$ hierarchical levels. Based on the aforementioned inference—that the failure of a large-scale structure is the cumulative result of failures in smaller structures to a certain degree—the number of magnitude $M$ earthquakes generated by the failure of level $n$ structures can be determined.

The number of earthquakes generated by structural failure, specifically the frequency of earthquakes of magnitude $M$ and above within a given period, represents the seismic count at a specific magnitude level. Assuming a relationship between the seismic magnitude $M$ and the hierarchical levels of the crustal structure, the correlation between earthquake frequency and magnitude reflects the spatial distribution of the local crustal architecture. Regarding the temporal aspect of seismicity—the relationship between the number of earthquakes and time—the logic of this paper suggests that it reflects a physical progressive process. In this process, the successive failure of small-scale crustal structures leads to the eventual failure of larger structures (i.e., the gradual accumulation of small earthquakes leading to a major seismic event).

Based on the hypothesis that the forces acting on various regions of the Earth's crust originate from the periodic influence of the mantle, and considering the long-term stability of the Earth's internal structure and orbital motion, this impact stress can be regarded as a specific constant for any given region. Under the influence of external stress, materials typically exhibit two primary evolutionary processes: the first is a strain process that enhances strength (such as work hardening in metals), and the second is a process that weakens strength through the destruction of the material structure (such as the fracturing process). For complex organizational structures like the Earth's crust, which are interwoven with plastic and brittle materials, the mechanisms for structural reinforcement and healing remain unclear. For the purposes of this study, we assume that any strengthening effects resulting from crustal failure are negligible during the subsequent phase of accelerated crustal fracturing, as the crustal integrity at that stage is insufficient to significantly influence the overall failure process. Since the failure of small structures degrades the overall regional strength, this leads to a compounding effect.

Under the influence of periodic constant stress, the structural failure of various crustal regions becomes an accelerated process, leading to subsequent stress impacts that generate even more structural damage. Consequently, it is posited that as a major earthquake approaches, the frequency of seismic events will accelerate over time. To theoretically describe the frequency of small earthquakes, a simplified approach is to assume that the rate of increase in the number of earthquakes of magnitude $M$ is proportional to the existing cumulative count of earthquakes at that specific magnitude.

𝑑𝑛 𝑖 0 𝑑𝑡 = p × 𝑛 𝑖 0 (5)

The relative frequency of earthquakes of different magnitudes, or their relative occurrence rate, does not vary over time. Consequently, within similar geological structures, this rate remains independent of the specific magnitude level.

𝑛 𝑖 0 = 𝑛 0 × 10 p ( 𝑡 − 𝑡 0 ) × 𝑙𝑔𝑒 (6) .

The number of earthquakes of a specific magnitude and the total number of occurrences within a given time period.

1 − 10 − 𝑏 = 10 𝑐 (8) .

and Crustal Structure

In the exponent of Equation (9), let $p(t - t_0) \times \lg e + c = a$; we then obtain:

According to the Gutenberg-Richter law (expressed as a function of time), for a specific seismic region, since the parameters are determined, there exists a definite cumulative frequency of earthquakes with a magnitude of $M$ or greater within a selected time interval.

𝑡 𝑖 = 𝑡 0 + \frac{i b - c}{p \lg e}

(11). This represents the magnitude of the first earthquake of a specific grade to occur in the region. When the magnitude of the potential earthquake is sufficiently small, it defines the recurrence period of major earthquakes within that region.

𝑡 𝑖 𝑚 − 𝑡 0 = \frac{i_m b - c}{p \lg e}

(12). Due to the existence of an upper limit on the magnitude of large regional earthquakes, the frequency of high-magnitude events increases as a major earthquake approaches. Consequently, by utilizing Equation (9), one can fit data from small earthquakes occurring prior to a major event in a specific region to obtain local seismic parameters. These parameters can then be used to calculate the local $t_i$ via the relevant formulas (thereby determining the time corresponding to the first $M_i$ in that area). During the fitting process, the small earthquake data can be predetermined; however, it is critical to note that when selecting seismic data, the size and location of the region must be carefully considered. The regional area should neither be too small nor too large. If the area is too small or poorly positioned, the small earthquake data will be incomplete, failing to capture the full process of a single large earthquake development at any given moment.

Conversely, if the area is too large, the data may originate from the preparation zones of two or more separate large earthquakes, thereby exceeding the scope of the theory's applicability. Furthermore, the time span of the selected data should be sufficiently long to ensure there is enough small earthquake data available to exclude interference from the aftershocks of previous large earthquakes in the region during the fitting process.

A feasible approach to determining the regional scope before an earthquake occurs is to segment the seismic data by region to identify potential focal areas. The advantage of this method is that it facilitates earthquake prediction based on formal equations. Specifically, it is assumed that the number of earthquakes of magnitude $M$ is related to the existing cumulative count of earthquakes of that magnitude. Under specific conditions that enhance the theoretical robustness, the number of earthquakes of magnitude $M$ can be considered a function of the rate of change in the cumulative count, expressed as:

𝑑𝑛 𝑖 0 𝑑𝑡 = p × 𝑛 𝑖 0 𝑞 (13).

𝑛 𝑖 0 = [ 𝑛 0 1 − 𝑞 + ( 1 − 𝑞 ) × p × ( 𝑡 − 𝑡 0 ) ]^{\frac{1}{1-q}}

(15). Relationship and Fitting

We can fix $i_0 = 0$ and $n_0 = 1$. In this case, $N_{i_0}(t_0) =$

$1 - 10^{-b}$. That is, at time $t = t_0$, the cumulative number of earthquakes of magnitude $i_0 = 0$ or greater is given by this value.

1 − 10 − 𝑏 × 10 − 𝑖𝑏 = 10 ( 𝑎 − 𝑖𝑏 ) (16) .

𝑎 = log { \frac{1 - 10^{-b}}{[1 + (1 - q) p (t - t_0)]^{\frac{1}{q-1}}} }

The first occurrence in this region...

𝑡 𝑖 = 𝑡 0 + \frac{10^{(i b - c)(q-1)} - 1}{(1 - q) p}

𝑇 𝑚 ≈ 𝑡 𝑖 𝑚 − 𝑡 0 = \frac{1}{(q - 1) p}

In cases where the series is large, the relationship between these variables can be analyzed. By applying the formula to fit small earthquake data occurring within a specific region prior to a major event, local seismic parameters—including $\beta$—can be obtained. Subsequently, these parameters can be utilized through the formula to estimate the time $t_f$ corresponding to the first occurrence of a future earthquake of magnitude $M$. Interestingly, the formula bears a striking resemblance to the Paris law, which describes the change in crack length $a$ during the fatigue of metallic materials. The Paris law characterizes the stable growth process of cracks under cyclic stress, a process that ultimately leads to the fracture of the material.

If we consider the magnitude of an earthquake to be equivalent to the crack length within the Earth's crust, and the cumulative number of earthquakes to be equivalent to the growth of that crack length, we can derive a similar relationship.

4.1 中国四川省某区域地震

A certain region in Sichuan Province, China.

The spatial and temporal distribution of seismicity in this region is non-uniform, and the number of major earthquakes currently in the nucleation stage remains unclear. In addition to analyzing the overall regional data, several sub-regions were analyzed independently. These sub-regions are denoted as specific study areas. To maintain consistency and minimize the impact of parameter fluctuations on the analytical results, certain variables were held constant in all subsequent fitting procedures.

Regarding the geographic distribution and the selection of analysis areas, the data fitting process was conducted by fixing the origin point. Specifically, the cumulative seismic magnitude was used as the primary metric for these calculations.

Prior to performing the data fitting, the expected...

∆ 𝑁 𝑖 ( 𝑡 ) = { [ 1 + ( 1 − 𝑞 ) × p × ( 𝑡 − 𝑡 0 ) ]^{\frac{1}{1-q}} - 1 } \times 10^{c - i b}

When fitting data, it is possible to obtain nearly identical results despite using different methodologies or model architectures. However, the underlying implications for generalization and physical interpretability can vary significantly. In the context of machine learning and statistical modeling, achieving a high degree of fit on a training dataset does not necessarily guarantee that the model has captured the true underlying distribution of the phenomena.

The phenomenon of obtaining similar numerical outputs from distinct models is often encountered in complex high-dimensional spaces. While the loss functions may converge to similar minima, the structural differences between models—such as the choice of activation functions in deep learning or the specific kernel in a support vector machine—can lead to divergent behavior when the models are applied to out-of-distribution data. Therefore, evaluating models solely based on their ability to fit existing data points can be misleading.

Furthermore, the stability of these results