Prognostic Properties of Instantaneous Amplitudes Maxima of Earth Surface Tremor
Abstract
:1. Introduction
2. Data
3. Principal Components of Increments in a Moving Time Window
4. Empirical Mode Decomposition
5. Ensemble Empirical Mode Decomposition
- Add a white noise implementation to the original data.
- Decomposition of data with the addition of white noise into empirical modes.
- Repeat steps 1 and 2 quite a large number of times with different implementations of white noise.
- Obtain the ensemble average for the corresponding empirical modes.
6. Hilbert Transform
7. Influence Matrix
- (1)
- a sequence of moments in time corresponding to the largest local maxima of the amplitudes of the envelopes at certain levels of the EEMD Huang decomposition
- (2)
- a sequence of times of seismic events with a magnitude not less than a given value.
8. Estimation of Connections between the Times of Local Amplitude Maxima and Seismic Events
- The minimum and maximum lengths of time windows and —the number of lengths of time windows in this interval are selected. Thus, the lengths of the time windows took on the values , , . In our calculations, we took as equal to 1 year, and —3 years, .
- Each time window of length was shifted from left to right along the time axis with some offset . Let us denote by , the sequence of moments in time of the positions of the right windows with length . The number of time windows in length is determined by their time offset . We used a time window offset of 0.01 year.
- For each position of a time window of length , the elements of the influence matrix (35) are estimated for a given relaxation time of the model (26–27), corresponding to the mutual influence of the two processes being analyzed. We took a value equal to 0.1 year. For definiteness, we will consider one influence, for example, of the first process on the second. As a result of such estimates, we obtain their values in the form , where is the corresponding element of the influence matrix for a position with a time window number of length .
- In the sequence , we select elements corresponding to local maxima of values , that is, from the condition . Let us present each element as a vertical segment of length located at a time point . The combination of such vertical graphic elements for all , visualizes the “strength” of the mutual influence of processes on each other.
9. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Clust#. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
Nsta | 57 | 56 | 54 | 83 | 69 | 61 | 78 | 77 | 91 | 76 | 57 | 95 | 48 | 88 | 57 |
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Lyubushin, A.; Rodionov, E. Prognostic Properties of Instantaneous Amplitudes Maxima of Earth Surface Tremor. Entropy 2024, 26, 710. https://doi.org/10.3390/e26080710
Lyubushin A, Rodionov E. Prognostic Properties of Instantaneous Amplitudes Maxima of Earth Surface Tremor. Entropy. 2024; 26(8):710. https://doi.org/10.3390/e26080710
Chicago/Turabian StyleLyubushin, Alexey, and Eugeny Rodionov. 2024. "Prognostic Properties of Instantaneous Amplitudes Maxima of Earth Surface Tremor" Entropy 26, no. 8: 710. https://doi.org/10.3390/e26080710