In this paper, some algorithms for constructing orthogonal and biorthogonal compactly supported wavelets on Vilenkin groups are suggested. As application, several examples of p-adic wavelets, which correspond to the refinable functions presented recently by the first author, are given.

A method is proposed for analyzing the tremor of the earth’s surface, measured by GPS, in order to highlight prognostic effects. The method is applied to the analysis of daily time series of vertical displacements in Japan. The network of 1047 stations is divided into 15 clusters. The Huang Empirical Mode Decomposition (EMD) is applied to the time series of the principal components from the clusters, with subsequent calculation of instantaneous amplitudes using the Hilbert transform. To ensure the stability of estimates of the waveforms of the EMD decomposition, 1000 independent additive realizations of white noise of limited amplitude were averaged before the Hilbert transform. Using a parametric model of the intensities of point processes, we analyze the connections between the instants of sequences of times of the largest local maxima of instantaneous amplitudes, averaged over the number of clusters and the times of earthquakes in the vicinity of Japan with minimum magnitude thresholds of 5.5 for the time interval 2012–2023. It is shown that the sequence of the largest local maxima of instantaneous amplitudes significantly more often precedes the moments of time of earthquakes (roughly speaking, has an “influence”) than the reverse “influence” of earthquakes on the maxima of amplitudes.

This study investigates the hypothesis that meteorological anomalies may precede earthquake events. Long-term time series of observations for air temperature, atmospheric pressure and precipitation at a meteorological station in Kamchatka are considered. Time series are subjected to Huang decomposition into sequences of levels of empirical oscillation modes (intrinsic mode functions—IMFs), forming a set of orthogonal components with decreasing average frequency. For each IMF level, the instantaneous amplitudes of envelopes are calculated using the Hilbert transform. A comparison with the earthquake sequence is made using a parametric model of the intensity of two interacting point processes, which allows one to quantitatively estimate the “measure of the lead” of the time instants of the compared sequences. For each IMF level, the number of time moments of the largest local maxima of instantaneous amplitudes which is equal to the number of earthquakes is selected. As a result of the analysis, it turned out that for the sixth IMF level (periods of 8–16 days), the “lead measure” of the instantaneous amplitude maxima of meteorological parameters in comparison with earthquake time moments significantly exceeds the inverse lead, which confirms the existence of prognostic changes in meteorological parameters in the problem of “atmosphere–lithosphere” interaction. This study reveals that certain meteorological anomalies can be a precursor for seismic activity.

Suppose that ω(φ, ·) is the dyadic modulus of continuity of a compactly supported function φ in L ²(ℝ⁺) satisfying a scaling equation with 2 ⁿ coefficients. Denote by α _φ the supremum for values of α > 0 such that the inequality ω(φ, 2^−j ) ≤ C2 ^−αj holds for all j ∈ ℕ. For the cases n = 3 and n = 4, we study the scaling functions φ generating multiresolution analyses in L ²(ℝ₊) and the exact values of α _φ are calculated for these functions. It is noted that the smoothness of the dyadic orthogonal wavelet in L ²(ℝ₊) corresponding to the scaling function φ coincides with α _φ .

An estimate of the trigger effect of the proton flux on seismicity was obtained. The proton flux time series with a time step of 5 min, 2000–2024, was analyzed. In each time interval of 5 days, statistics of the proton flux time series were calculated: mean values, logarithm of kurtosis, spectral slope, singularities spectrum support width, wavelet-based entropy, and the Donoho–Johnston wavelet-based index. For each of the used statistics, time points of local extrema were found, and for each pair of time sequences of proton flux statistics and earthquakes with a magnitude of at least 6.5 in sliding time windows, the “advance measures” of each time sequence relative to the other were estimated using a model of the intensity of interacting point processes. The difference between the “direct” measure of the advance of time points of local extrema of proton flux statistics relative to the time moments of earthquakes and the “inverse” measure of the advance was calculated. The maximum proportion of the intensity of seismic events for which the proton flux was a trigger was estimated as 0.28 for using the points of the local minima of the singularities spectrum support width.

—The approximation neural network (ANN) method for solving the inverse geoelectric problem in the piecewise constant classes of media with the subsequent refinement of the model by random search method is presented. A conditionally well-posed magnetotelluric inverse problem is considered on a regularized parameterization grid. The total number of the sought-for parameters in this problem can be \(\sim n \times {\text{1}}{{0}^{{\text{3}}}}\). The work of the algorithm is illustrated by the example of 3D inversion of the field survey data. The posterior ambiguity characteristics of the obtained solution were calculated. The proposed methods are implemented using supercomputing cluster resources and parallel computing technology.