The Relationship Between the Seismic Regime and Low-Frequency Variations in Meteorological Parameters Measured at a Network of Stations in Japan

Abstract

The relationship between the seismic regime and humidity, pressure, temperature, and wind speed measured at a network of stations on the Japanese islands was studied for the period from 1973 to 2025. For each of the parameters, weighted average time series were constructed using the principal component method and then subjected to wavelet decomposition. For wavelet decomposition levels, the amplitudes of the envelopes and the points of their local extrema were found and compared with the times at which earthquakes occurred. The problem of estimating an advanced measure of envelope extremum points relative to earthquake moments was considered using a model of interacting point processes. For a sequence of 213 strong earthquakes with a magnitude of at least 6.5, the same numbers for the largest local maxima and the smallest local minima were selected for the extrema of the envelope amplitude of each parameter. It turned out that the largest advance measures occurred for the seventh level of decomposition (the period from 16 to 32 days). Two advance mechanisms were identified: one mechanism is associated with the trigger effect of cyclones on seismicity, and the second is associated with the occurrence of atmospheric earthquake precursors.

1. Introduction

Changes in meteorological parameters associated with imminent earthquakes or occurring after seismic events have long been the subject of intensive research. These changes involve not only the atmosphere but also the electromagnetic fields in the ionosphere. In recent decades, these studies have led to the emergence of “new non-seismic precursors” that differ from traditional precursors associated with changes in the seismic regime, such as variations in the slope of the recurrence graph, and are oriented towards extracting information almost exclusively from seismic catalogues [1]. These precursors are considered to be a result of the interaction of many processes included in the global electric circuit, affecting air ionization and changes in the thermodynamic parameters of the atmosphere [2,3]. The general concept of lithosphere–atmosphere–ionosphere interaction (LAIC) for the formation of short-term precursors as a result of interactions among processes in the Earth’s atmosphere, ionosphere, and magnetosphere in the final stages of earthquake development was considered in [4,5]. From the point of view of the formation of precursors in the atmosphere, the key factor in these works is proposed to be the ionization of air and the formation of water condensation nuclei by ions because of the action of radioactive radon on atmospheric gases. Radon emanation from the Earth’s interior occurs through the boundaries of the Earth’s crust blocks, including through active underwater tectonic faults. The application of this concept to the analysis of earthquake precursors was considered in [6,7]. In [8,9,10], methods for analyzing radon emanation data (time series and spatial distribution) before strong earthquakes and for assessing potentially dangerous seismic areas were proposed based on the use of fractal statistics and power distribution laws.

Changes in the atmosphere are closely related to processes in the ionosphere. The authors of [11,12,13] studied the electron density of the ionosphere, the formation of the distribution of electric potentials in the ionosphere, and the features of electromagnetic ULF/ELF radiation before strong earthquakes, including the Tohoku mega-earthquake on 11 March 2011, in Japan. In [14], multiparameter observations before two strong earthquakes in Japan were analyzed from the point of view of the occurrence of low-frequency changes in the magnetic field, the total electron content, and synchronous anomalous changes in meteorological parameters (temperature and humidity), which could be interpreted as precursors of seismic events. Similar studies were presented in [15,16], using observations from before a series of earthquakes in New Zealand and Greece. In [17], using spectral and wavelet analysis, the occurrence of an atmospheric gravity wave before an earthquake (M = 6.7) in India was studied. The occurrence of short-term precursors in China in the form of an anomalous vertical electric field of the atmosphere due to the ionization of decay products of radioactive radon at different altitudes was considered in [18]. In [19], it was shown that in the preparation zone of a large earthquake in Mexico (M = 7.4), variations in the temperature and relative humidity of atmospheric air reached peak values within 10 days before the upcoming earthquake due to later disturbances of the total electron content in the ionosphere.

The occurrence of electrical impulses in the atmosphere in the vicinity of seismically active faults, with subsequent air ionization and the formation of aerosols and ascending air currents, was studied in [20]. Variations in the total electron content in the ionosphere before an earthquake in Haiti (M = 7) were analyzed in [21] using parametric time series models and neural networks. The authors of [22] showed that some climatological anomalies may appear in the epicentral region weeks before large earthquakes. In particular, for several earthquakes in Central Italy, anomalies in temperature, water vapor, and ozone content occurred within 2 months before the seismic events. The authors of [23] presented the results of detecting anomalies in air temperature, relative humidity, and pressure that occurred one day before a seismic event in China, while [24] showed that disturbances in atmospheric circulation in the form of interruptions of climatic cyclone trajectories were a trigger for the occurrence of earthquake swarms in Iran. Similar studies on atmospheric processes such as abnormally low pressure, showers, storms, and thunderstorms as triggers provoking earthquakes in Japan were presented in [25].

The influence of the groundwater level and observed precipitation in the area of swarming earthquakes in southeastern Germany on the migration of hypocenters, determined using the pore pressure diffusion model, was presented in [26]. The problem of determining the most informative atmospheric and geochemical precursors in Japan for earthquakes with a magnitude of at least 6 was considered in [27]. The possibility of constructing a short-term forecast based on a model of interaction between the lithosphere, atmosphere, and ionosphere was investigated in [28] using the example of a retrospective analysis of atmospheric parameters before a strong earthquake (M = 7.8) in Mexico. The selection of short-term precursors based on anomalous behavior of the temperature, relative humidity, and atmospheric chemical potential was studied in [29,30], including using the natural time apparatus to select critical behavior [29]. Similar studies were presented in [31] using the example of analyzing the processes of preparation for a strong seismic event in Altai. The use of satellite-based surface latent heat flux observations to extract the precursor behavior of large earthquakes (M = 7.8, 7.3, 7.0) in Nepal and Japan using the natural time approach was presented in [32]. The analysis of surface latent heat flux together with atmospheric chemical potential for retrospective prediction of an earthquake in Crete, Greece, was presented in [33]. Multiparameter precursors of different physical nature describing the ionospheric and atmospheric states associated with a seismic event in Japan (M = 7.1) were analyzed in [34] using several types of neural networks.

This work continues the study carried out in [35], which analyzed the relationships between anomalies in meteorological parameters (pressure, temperature, and precipitation) and the seismic regime on the Kamchatka Peninsula; the parameters were recorded at 3 h intervals over a long period of time, from 4 November 1993 to 30 September 2024, at the Pionerskaya meteorological station in Kamchatka. In this study, the analysis was performed not for one station but for a network of meteorological stations on the Japanese islands and for a recording time of 52.5 years, from the beginning of 1973 to mid-2025.

2. Initial Data

The temperature, humidity, barometric pressure, and wind speed time series used in this study are freely available from https://www.ncei.noaa.gov/data/global-hourly/archive/csv/ (accessed on 20 July 2025). The numbers of meteorological stations in Japan with available temperature, humidity, barometric pressure, and wind speed data are 386, 379, 251, and 387, respectively. The earliest start of the time series dates back to 1946, but there are significant gaps in the data before 1973. Since 1973, the number of stations with available data has increased sharply. Therefore, time series starting from 1973 with a longest gap duration not exceeding 0.5 years were further selected. The gaps were filled based on the behavior of the data to the left and right of the gap time interval in time segments of length equal to the gap length. The time series were then reduced to a uniform time step of 3 h by averaging the data within successive 3 h time intervals. As a result, time series of humidity from 111 stations, temperature from 112 stations, wind speed from 112 stations, and atmospheric pressure from 53 stations were obtained. The length of the obtained synchronous time series was 153,452 readings at intervals of 3 h from the beginning of 1973 to 7 July 2025 (52.5 years).

For each meteorological parameter, the time series of the weighted average was calculated using the principal component method [36], in which the squares of the components of the eigenvector corresponding to the maximum eigenvalue of the correlation matrix of the multidimensional time series were taken as weights. By construction, the sum of the weights was equal to one. A station’s weight reflects the contribution of measurements made at that station to the overall average. In Figure 1, the positions of the meteorological stations for which the time series for humidity, temperature, wind speed, and atmospheric pressure were used are shown with colored dots. The color of the dot reflects the rank of the station weight, which contributes to the calculation of the weighted average. A total of 6 ranks were used; to calculate them, the range of change in weights for each parameter was divided into 6 intervals containing the same number of elements. On the maps shown in Figure 1, the remote station MINAMI TORISHIMA (Lat = 24.289697, Lon = 153.979119) stands out; its weight for all parameters belonged to the lowest rank.

Figure 2 shows the graphs of the average weighted time series of humidity, atmospheric pressure, temperature, and wind speed. Once again, we note that these graphs represent one of the variants of the first principal component of the original multivariate time series.

3. Local Extremum Points of Envelopes After Wavelet Filtering

Each of the time series presented in Figure 2 was then subjected to decomposition into orthogonal components using a smooth 20th-order symlet with 10 vanishing moments [37]. The choice of the optimal orthogonal wavelet was made based on the principle of minimum entropy of the distribution of the squared wavelet coefficients of all orthogonal Daubechies wavelets with numbers of vanishing moments from 1 to 10.

The minimum period corresponding to level $k$ of the details of the wavelet decomposition is $\mathrm{\Delta }t\cdot 2^k$, where $\mathrm{\Delta }t$ is the time step of the time series. The maximum period of level $k$ is $\mathrm{\Delta }t\cdot 2^{k+1}$. Since the time step is 3 h, the range of periods for the seventh level varies from 384 to 768 h, i.e., from 16 to 32 days.

For wavelet decompositions at the seventh level of detail, the graphs of which are presented in Figure 3, their envelopes were calculated using the Hilbert transform [38].

Figure 4a shows the time sequence of seismic events with a magnitude of at least 6.5 in the rectangular vicinity of the Japanese islands, shown in Figure 1, for the time interval from the beginning of 1973 to 7 July 2025. Figure 4a–d show the envelopes for the seventh level of wavelet decomposition of the weighted average time series of humidity, pressure, temperature, and wind speed and indicate the positions of the 213 most expressive local extrema of the envelopes: red dots mark local maxima, and blue dots mark local minima.

4. Measures of Advance of Local Extrema of Envelope Time Moments with Respect to Earthquake Occurrence

In the following, we are interested in the question of whether there is an advance of the times of the largest local maxima or smallest local minima of the envelopes for the seventh level of wavelet decompositions, shown in Figure 4, with respect to the occurrence times of earthquakes. In this case, we separately consider the pairs of event sequences in Figure 4a and the largest local maxima of the envelopes for the seventh level of wavelet decompositions (red dots in Figure 4b–e) and the smallest local minima of the envelopes (blue dots in Figure 4b–e). In total, we consider four pairs of time sequences, in which one of the sequences will always be the sequence of earthquakes with a minimum magnitude of 6.5, shown in Figure 4a. Clarifying these issues also requires estimating the “reverse” advance and calculating their difference. If the average value of this difference is positive, then there is a trigger effect of the proton flux on seismicity. In addition, the average difference in the lead measures will give a measure of the trigger effect.

To clarify these issues, we apply the influence matrix method, which was previously used in [35,39,40,41] to analyze the relationships between the occurrence times of seismic events and the time points of local extremes of various statistics from meteorological time series, magnetic field fluctuations, Earth surface tremors, and the influence of proton flux on the seismic process.

Let $t_j^{(\alpha )}, j=1,\cdots ,N_{\alpha }; \alpha =1,2$, represent the moments of time of two event streams. In our case, these are as follows:

(1) A sequence of moments of time corresponding to the points of the most “expressive” local extrema of the envelopes of wavelet decompositions at the seventh level of detail;

(2) A sequence of moments of time at which earthquakes with a magnitude of at least 6.5 occurred.

Let us represent their intensities as follows:

$${\lambda }^{(\alpha )}(t)=b_0^{(\alpha )}+b_1^{(\alpha )}{g}^{(1)}(t)+b_2^{(\alpha )}{g}^{(2)}(t)$$

(1)

where $b_0^{(\alpha )}\ge 0,b_{\beta }^{(\alpha )}\ge 0$ are the parameters, and $g^{(\beta )}(t)$ is the influence function of events $t_j^{(\beta )}$ of flow $\beta$:

$$g^{(\beta )}(t)={\displaystyle {\sum }_{t_j^{(\beta )}<t}\exp (-(t-t_j^{(\beta )})/\tau )}$$

(2)

According to Formula (2), the weight of event $j$ becomes non-zero for times $t>t_j^{(\beta )}$ and decays with characteristic time $\tau$. The parameter $b_{\beta }^{(\alpha )}$ determines the degree of influence of event flow $\beta$ on flow $\alpha$. The parameter $b_{\alpha }^{(\alpha )}$ determines the degree of influence of flow $\alpha$ on itself (self-excitation), and the parameter $b_0^{(\alpha )}$ reflects a purely random (Poisson) component of intensity. Let us fix the parameter $\tau$ and consider the problem of determining the parameters $b_0^{(\alpha )},b_{\beta }^{(\alpha )}.$

The log-likelihood function for a non-stationary Poisson process is equal to the following over a time interval [42]:

$$\ln (L_{\alpha })={\displaystyle \sum _{j=1}^{N_{\alpha }}\ln ({\lambda }^{(\alpha )}(t_j^{(\alpha )}))-{\displaystyle \underset{0}{\overset{T}{\int }}{\lambda }^{(\alpha )}(s)ds}},\hspace{1em}\alpha =1,2$$

(3)

It is necessary to find the maximum of function (3) with respect to parameters $b_0^{(\alpha )},b_{\beta }^{(\alpha )}$. It can be shown that this problem reduces to solving the following maximization problem (for the derivation details, see [35]):

$${\displaystyle \sum _{j=1}^{N_{\alpha }}\ln ({\lambda }_0^{(\alpha )}+b_1^{(\alpha )}\mathrm{\Delta }{g}^{(1)}(t_j^{(\alpha )})+b_2^{(\alpha )}\mathrm{\Delta }{g}^{(2)}(t_j^{(\alpha )}))\to \underset{b_1^{(\alpha )},b_2^{(\alpha )}}{\max }}$$

(4)

under the conditions

$$b_1^{(\alpha )}\ge 0,\hspace{1em}{b}_2^{(\alpha )}\ge 0,\hspace{1em}{b}_1^{(\alpha )}{\overline{g}}^{(1)}+b_2^{(\alpha )}{\overline{g}}^{(2)}\le {\lambda }_0^{(\alpha )}$$

(5)

Here, ${\lambda }_0^{(\alpha )}=N_{\alpha }/T$, and $\mathrm{\Delta }{g}^{(\beta )}(t)=g^{(\beta )}(t)-{\displaystyle {\int }_0^T{g}^{(\beta )}(s)ds}/T$.

Having solved problems (4) and (5) numerically for a given $\tau$, we can enter the elements of the influence matrix ${\kappa }_{\beta }^{(\alpha )},\alpha =1,2;\beta =0,1,2$, according to the following formulas:

$${\kappa }_0^{(\alpha )}=b_0^{(\alpha )}/{\lambda }_0^{(\alpha )}\ge 0,\hspace{1em}{\kappa }_{\beta }^{(\alpha )}=b_{\beta }^{(\alpha )}\cdot {\overline{g}}^{(\beta )}/{\lambda }_0^{(\alpha )}\ge 0$$

(6)

We can then determine the influence matrix:

$$\left(\begin{array}{l}{\kappa }_0^{(1)}{\kappa }_1^{(1)}{\kappa }_2^{(1)}\\ {\kappa }_0^{(2)}{\kappa }_1^{(2)}{\kappa }_2^{(2)}\end{array}\right)$$

(7)

The first column of matrix (7) is composed of Poisson fractions of mean intensities. The diagonal elements of the right submatrix of size 2 × 2 consist of self-excited elements of mean intensity, while the off-diagonal elements correspond to mutual excitation. The sums of the component rows of influence matrix (7) are equal to 1. The influence matrices are estimated in a certain sliding time window of length $L$ with offset $\mathrm{\Delta }L$ and with a given value of the attenuation parameter $\tau$.

When analyzing variations in the components of influence matrices in sliding time windows corresponding to the mutual influence of the analyzed time sequences, the main attention is paid to their local maxima with their subsequent averaging. Below is described a method for processing these local maxima and averaging them for a set of time window lengths changing within specified limits.

(1) The minimum $L_{\min }$ and maximum $L_{\max }$ lengths of time windows are selected, and $N_L$ denotes the number of lengths of time windows in this interval. Thus, the lengths of the time windows take the values $L_k=L_{\min }+(k-1)\mathrm{\Delta }L$, $k=1,\cdots ,N_L$, $\mathrm{\Delta }L=(L_{\max }-L_{\min })/(N_L-1)$, and $N_L=50$.

(2) Each time window of length $L_k$ is shifted from left to right along the time axis with some offset $\mathrm{\Delta }t$. Let $t_j(L_k)$, $j=1,\cdots ,M(L_k)$, denote the sequence of time moments of the positions of the right windows of length $L_k$. The number $M(L_k)$ of time windows of length $L_k$ is determined by their time offset, $\mathrm{\Delta }t$. We used a time window offset $\mathrm{\Delta }t$ of 0.2 years.

(3) For each position of the time window of length $L_k$, the elements of influence matrix (11) are estimated for a given relaxation time $\tau$ of models (1) and (2), corresponding to the mutual influence of the two analyzed processes. For definiteness, we consider some influence, for example, of the first process on the second. As a result of such estimates, we obtain their values in the form $(t_j(L_k),c_j(L_k))$, where $c_j(L_k)$ is the corresponding element of the influence matrix for position $j$ of the time window of length $L_k$.

(4) In the sequence $(t_j(L_k),c_j(L_k))$, we select elements $(t_j^{*}(L_k),c_j^{*}(L_k))$ corresponding to local maxima of values $c_j(L_k)$, that is, from the condition $c_{j-1}(L_k)<c_j^{*}(L_k)<c_{j+1}(L_k)$.

(5) We select a “small” time interval of length $\epsilon$, and for a sequence of such time fragments, we calculate the average value of local maxima $c_j^{*}(L_k)$ for which their time marks $t_j^{*}(L_k)$ belong to these fragments. Averaging is performed over all lengths of time windows $L_k,k=1,\cdots ,N_L$. In our calculations, we took length $\epsilon$ equal to 0.1 years.

5. Optimal Selection of Parameters and Two Advance Mechanisms

The complete set of free parameters for the influence matrix method is $\tau$, $L_{\min }$, $L_{\max }$, $N_L$, $\mathrm{\Delta }t$, and $\epsilon$. The most critical for the calculation results is the choice of parameters $\tau$, $L_{\min }$, and $L_{\max }$. For each set of values $(\tau ,L_{\min },L_{\max })$, it is possible to calculate the average value of the lead measure $G(T_{extr}\to EQ)$ between the moments of time $T_{extr}$ of the most expressive local extrema of the envelopes relative to the moments of time of earthquakes and the average value $G(EQ\to T_{extr})$ of the inverse lead measure. We are interested in the difference in the average measures: $\mathrm{\Delta }G(\tau ,L_{\min },L_{\max })=G(T_{extr}\to EQ)-G(EQ\to T_{extr})$. The value $\mathrm{\Delta }G$ is the averaged value of the difference ${\kappa }_2^{(1)}-{\kappa }_1^{(2)}$ of elements of matrix (7). The choice of parameters $(\tau ,L_{\min },L_{\max })$ was made in accordance with the following conditions:

$$\mathrm{\Delta }G(\tau ,L_{\min },L_{\max })\to \max ,\hspace{1em}{\tau }_{\min }\le \tau \le {\tau }_{\max },\hspace{1em}{L}_{\min }^{(\min )}\le L_{\min }\le L_{\min }^{(\max )},\hspace{1em}{L}_{\max }^{(\min )}\le L_{\max }\le L_{\max }^{(\max )}$$

(8)

The a priori limits of parameter variation in problem (8) were as follows (in years): ${\tau }_{\min }=0.05$, ${\tau }_{\max }=10$, $L_{\min }^{(\min )}=1$, $L_{\min }^{(\max )}=3$, $L_{\max }^{(\min )}=4$, and $L_{\max }^{(\max )}=10$. Problem (8) was solved numerically using the random enumeration method, allowing parallel implementation of many independent search flows (in total, 3000 random trials for each variant) with the final selection of the best result. In this method, the maximum point of function (8) was found by probing a three-dimensional parallelepiped $(\tau ,L_{\min },L_{\max })$ of possible parameters with random values.

As a result of calculating all the differences $\mathrm{\Delta }G$ in the average lead measures for the optimal parameters found from the solution of problem (8), for the variants of sets of pairs of time moment sequences, the sequences of extremum points with maximum differences were selected. The results of such a choice are presented in

Table 1. Optimal parameters of point processes interaction model.

Property	$\mathit{\tau }$	${\mathit{L}}_{\mathbf{min}}$	${\mathit{L}}_{\mathbf{max}}$	$\mathbf{\Delta }\mathit{G}$
Humidity, Max	9.74	2.84	7.59	0.102
Humidity, Min	0.98	2.60	4.00	0.228
Pressure, Max	9.30	2.92	7.97	0.140
Pressure, Min	0.98	2.81	8.04	0.455
Temperature, Max	9.78	2.52	8.65	0.100
Temperature, Min	0.78	2.99	4.69	0.255
Wind Speed, Max	0.85	2.93	5.92	0.303
Wind Speed, Min	0.49	2.97	4.11	0.217

, which displays the optimal values of the parameters $(\tau ,L_{\min },L_{\max })$ in years for all variants of the considered pairs of data and the corresponding maximum values of the difference $\mathrm{\Delta }G$.

The results of solving problem (8) can be divided into two groups. The first group is a set of parameters of models (1) and (2), for which the value $\mathrm{\Delta }G$ is maximum for each pair of local extremum points corresponding to each of the four meteorological quantities. This group corresponds to the points of local minima of humidity, atmospheric pressure, and temperature and the points of local maxima of wind speed. In Table 1, this group of parameters is highlighted in blue. The second group of parameters is made up of the remaining rows of Table 1, which are marked in black. This second group corresponds to the points of local maxima of humidity, pressure, and temperature and the points of local minima of wind speed.

The main difference between the first group (blue color) and the second (black color) is the difference in the optimal values of the relaxation parameter $\tau$ found from the solution of problem (8) for humidity, pressure, and temperature. In the first group, these values are 0.98, 0.98, and 0.78, while in the second group, they are an order of magnitude greater: 9.74, 9.30, and 9.78.

Such a strong difference suggests that there are two physically different mechanisms of advancement. We hypothesize that the first advance with small relaxation times (blue words in Table 1) is related to the trigger mechanism of cyclones’ influence on seismic processes, since cyclones are characterized by low pressure, a drop in temperature, and increased wind. As for the second mechanism, with large relaxation times (black words in Table 1), we assume that it is related to the formation of atmospheric earthquake precursors [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34], which are characterized by an increase in humidity and temperature as a result of the effect of radioactive radon on atmospheric gases and air ionization.

In the following, we denote these two mechanisms as the trigger and precursor advances. As can be seen from Table 1, the trigger’s atmospheric effect exceeds the precursor’s, which is one of the reasons for the difficulties in detecting atmospheric precursors in practice.

Figure 5 shows graphs of the changes in the lead measures for all variants from Table 1. The left-hand column of graphs (blue) corresponds to the set of parameters from Table 1 that relate to the trigger mechanism of the lead, and the right-hand column of graphs (black) relates to the mechanism of occurrence of atmospheric precursors.

6. Average Lead Measures

To highlight the general properties of change in the lead measures separately for the trigger and precursor mechanisms, we averaged the curves in the left- and right-hand columns of Figure 5 in successive time fragments 0.1 years long and compared them with a graph of the logarithm of the released seismic energy, calculated in a sliding time window of 1 year with a shift of 0.1 years for all earthquakes. The results of such calculations are presented in Figure 6.

It is of interest to estimate the extent to which the curves of change in the average lead measures in Figure 6b,c themselves additionally lead the seismic energy emissions shown in Figure 6a. To do this, we calculated the correlation coefficients between the values $LgE(t)$ of the logarithm of the discharged seismic energy (Figure 6a) and the average lead measure $C(t)$ (Figure 6b,c), where $t$ is the discrete time index. We calculated the correlation coefficients between $C(t)$ and $LgE(t+k)$ with time shift $k$ changing within certain limits $-k_{\max }\le k\le k_{\max }$, that is, the correlation function. Since the correlated values in Figure 6 differed greatly from Gaussian ones, we used the Spearman rank correlation coefficient [43]. Figure 6a shows a graph of the Spearman correlation function $<C(t)\cdot LgE(t+k)>$, where the angle brackets indicate calculation of the correlation coefficient. The limits of change in the time shift $k$ were from −3 to +3 years. Figure 7a shows that the correlation functions are strongly asymmetric and shifted towards negative $k$ values, which means that the variations in the average measures of advancement of changes in the release of seismic energy are ahead.

In addition, Figure 7b presents the estimated power spectra of curves of change in the average lead measures and indicates the periods corresponding to the spectral peaks. The spectral peak with a period of 12.8 years, which is close to the period of solar activity, is noteworthy. This spectral peak is most strongly expressed for the average lead measure in Figure 6c, corresponding to the second lead mechanism with large values of the relaxation parameter $\tau$, presumably associated with the formation of atmospheric precursors.

7. Estimates for the First and Second Parts of Data

The presented results were obtained by solving problem (8) for all available data. We further investigated the variability of lead measures by estimating the model parameters separately for the first and second halves of the data. The date 3 April 1999, was used to divide the sample into two equal parts. In the first half of the time interval, the number of earthquakes with a magnitude of at least 6.5 was 106, and in the second half, this number was 107. We repeated the entire sequence of calculations separately for the first and second halves of the data; that is, we calculated the weighted means, took the seventh level of detail of their wavelet decomposition, calculated the envelope amplitudes, and took the 106 most prominent local extrema of the envelope amplitudes for the first half and 107 local extrema for the second half. Next, by solving problem (8) separately for the first and second halves of the data, we found the optimal values of the parameters $(\tau ,L_{\min },L_{\max })$. The obtained values are given in

Table 2. Optimal parameters for the data before 1999.04.03.

Property	$\mathit{\tau }$	${\mathit{L}}_{\mathbf{min}}$	${\mathit{L}}_{\mathbf{max}}$	$\mathbf{\Delta }\mathit{G}$
Humidity, Max	6.02	3.00	9.14	0.262
Humidity, Min	1.07	2.71	4.43	0.158
Pressure, Max	8.11	2.94	9.70	0.243
Pressure, Min	0.95	2.83	7.93	0.628
Temperature, Max	8.49	2.97	9.40	0.248
Temperature, Min	9.43	2.68	4.04	0.121
Wind Speed, Max	4.93	2.91	9.67	0.283
Wind Speed, Min	2.49	2.81	8.72	0.323

and

Table 3. Optimal parameters for the data after 1999.04.03.

Property	$\mathit{\tau }$	${\mathit{L}}_{\mathbf{min}}$	${\mathit{L}}_{\mathbf{max}}$	$\mathbf{\Delta }\mathit{G}$
Humidity, Max	0.14	2.88	6.88	0.267
Humidity, Min	0.51	2.54	4.11	0.338
Pressure, Max	0.42	2.11	4.33	0.207
Pressure, Min	0.24	1.86	6.11	0.202
Temperature, Max	1.27	2.77	8.64	0.218
Temperature, Min	0.60	2.72	9.81	0.094
Wind Speed, Max	1.92	2.93	6.20	0.123
Wind Speed, Min	0.07	1.80	4.53	0.018

.

Figure 8 shows graphs of the lead measures for the parameters from Table 2 and Table 3.

Figure 8 shows the graphs of the lead measures for the parameters from Table 2 and Table 3. It is evident that for some meteorological quantities, the result changed dramatically compared with the model parameter estimate for the entire observation period. In particular, the local minima of the wind speed envelope amplitudes for the second half of the data (Figure 8(h2)) have a nearly zero lead measure relative to seismic event times. At the same time, the local minima of the pressure envelope amplitudes for the first half of the data demonstrate an exceptionally high lead measure (Figure 8(b1)). In the second half of the data, the lead measure of the local maxima of the humidity envelope amplitude shows a sharp surge in values for the 2002–2025 period (Figure 8(e2)).

Estimating the parameters of the point process interaction model separately for the first and second halves of the data shows that the obtained optimal values are highly dependent on the completeness of the sample. To obtain the most statistically significant results, it is important to use as representative a data set as possible.

8. Discussion

In earthquake forecasting, it is important to identify reliable short-term precursors of seismic events. Recently, much hope has been placed on the detection of reliable atmospheric–ionospheric anomalies before earthquakes, identified primarily using satellites [1,44]. This work was devoted to the analysis of long-term meteorological data obtained from a network of ground-based stations. Experience in analyzing such data, previously conducted in [35] and in this study, shows that meteorological precursors of stationary observations are located in periods in the range of 8–32 days. Therefore, their use for short-term forecasting is problematic. The goal of this study was to identify low-frequency relationships between processes in the atmosphere and the Earth’s crust.

A significant difference in the optimal parameters of the interacting event sequence model is noticeable when considering the entire sample (Table 1) and separately for each of the halves (Table 2 and Table 3). This means that the used model is very sensitive to the sample size. This feature is related to the solution of problem (8), for which the behavior of the envelope amplitudes over the entire time interval under consideration is important. If the parameter $(\tau ,L_{\min },L_{\max })$ values are fixed and the lead measures are estimated for them, the behavior of the curves in Figure 5 and Figure 8 will be almost identical. However, this raises the problem of choosing these parameters. Selecting parameters based on the condition of maximizing the difference $\mathrm{\Delta }G$ between the average lead curves allows us to identify two lead formation mechanisms, but also makes the problem unstable. We believe that for the most reliable identification of the various lead mechanisms, problem (8) should be solved for the entire sample.

As it turns out, pressure minima in the 16–32-day period range exhibit a very large lead time relative to earthquake times (Figure 5(b1) and Figure 8(b1)). We interpret this effect as a manifestation of the triggering effect of cyclones on seismicity. Regarding atmospheric precursors, humidity and temperature maxima exhibited a large lead time within the period range under consideration. If we consider estimation of the event sequence interaction model parameters as a kind of learning from past data, then the effects of a sharp increase in the lead time, such as that shown in Figure 8(e2), can be used for short-term forecasting.

In addition, the analysis of long-term meteorological data makes it possible to identify very-low-frequency features of interaction between the atmosphere and the lithosphere, such as a spectral peak with a period of 12.8 years for the average lead time for atmospheric precursors (Figure 7b).

9. Conclusions

A method was proposed herein for analyzing several long-term meteorological time series measured at a network of stations to compare their anomalies with the seismic regime. The data processing was based on the sequential application of the principal component method, wavelet decomposition, the calculation of envelopes using the Hilbert transform, and the application of a parametric model of coupled point processes to local extrema of the envelope amplitudes and the sequence of seismic events.

The method was illustrated by an example analysis of data from a network of meteorological stations in Japan for the period from early 1973 to mid-2025. As a result of the analysis, two mechanisms were discovered for the advance of local extremum points of envelope amplitudes from the wavelet decomposition of humidity, pressure, temperature, and wind speed time series at the low-frequency seventh level of detail (periods from 16 to 32 days) relative to the occurrence times of earthquakes with a magnitude of at least 6.5. The mechanisms differ by an order of magnitude in the optimal relaxation times of the models of interacting point processes.

The first mechanism, which was classified as triggering, is characterized by the advance of local minimum humidity, pressure, and temperature points and local maximum wind speed points relative to the time of seismic events. Its cause is the triggering effect of cyclones on the occurrence of earthquakes.

The second mechanism, which was classified as a precursor, is characterized by the advance of local maxima of humidity, pressure, and temperature and local minima of wind speed relative to the time of seismic events. Its cause is the occurrence of atmospheric precursors because of the impact of radioactive radon, released from faults in the Earth’s crust before earthquakes, on atmospheric gases and the subsequent ionization of air.

The averaged lead curves for both mechanisms also lead to the seismic energy release calculated for all earthquakes, as follows from the asymmetry of the Spearman correlation function.

For the averaged curve of the precursor-type lead measure, strong periodicity with a period close to that of solar activity was discovered.