# Adaptive Approach to Time-Frequency Analysis of AE Signals of Rocks

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Adaptive Matching Pursuit

- ${R}_{0}\left(t\right)=x\left(t\right)$, $i=0$.
- Set grid step for each parameter as half of the sampling interval of this parameter $\lambda =\{0.5{\lambda}_{\mathrm{max}},0.5{\lambda}_{\mathsf{\Delta}},0.5{\lambda}_{\mathrm{end}},0.5{\lambda}_{f}\}$.
- In the neighborhood of ${\mathbf{p}}_{j}$, construct a new grid, containing three points for each parameter p: $p-\lambda ,p,p+\lambda $.
- Define atom with the biggest correlation with ${R}_{i}\left(t\right)$ among the atoms corresponding to the grid nodes. ${\mathbf{p}}_{j+1}$ is the atom parameters.
- If $\lambda $ is less than a minimum possible step (corresponds to selected accuracy) then go to step 9.
- If ${\mathbf{p}}_{j+1}={\mathbf{p}}_{j}$ then decrease the step $\lambda =\lambda /2$.
- $j=j+1$, repeat from step 4.
- Estimate the residual signal ${R}_{i+1}\left(t\right)$ using ${g}_{k}(t,{\mathbf{p}}_{j})$.
- If the ratio of the residual signal norm to the initial signal norm becomes less than the prescribed threshold $\epsilon $ ($\parallel {R}_{i}\left(t\right)\parallel /\parallel x\left(t\right)\parallel <\epsilon $) then STOP, otherwise $i=i+1$ and repeat from step 2.

## 3. Description of the Experiment

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**AE sensors: (

**a**) hydrophone with integrated amplifier, (

**b**) system of directed hydrophones, (

**c**) combined receiver.

**Figure 2.**Effect of the parameters ${T}_{\mathrm{end}}$, ${p}_{\mathrm{max}}$ and $\mathsf{\Delta}$ on the form of functions: (

**a**) Gaussian function, (

**b**) Berlage function.

**Figure 4.**Example of geoacoustic pulse decomposition. (

**a**) Geoacoustic pulse, (

**b**) the first three atoms (each atom is marked by a different color) included into pulse decomposition, (

**c**) graph of value $\parallel {R}_{i}\left(t\right)\parallel /\parallel x\left(t\right)\parallel $ fall.

**Figure 5.**Kamchatka peninsula map with locations of epicenters of earthquakes considered in the paper. Blue triangle (▴) indicates Mikizha lake, red circles (•) are earthquake epicenters.

**Figure 6.**Dynamics of the probability density of distribution of the most powerful frequencies of geoacoustic pulses before and after the earthquake No. 1. $p\left(f\right)$ is the frequency density.

**Figure 7.**Dynamics of the probability density of distribution of the most powerful frequencies of geoacoustic pulses before and after the earthquake No. 2.

**Figure 8.**Dynamics of the probability density of distribution of the most powerful frequencies of geoacoustic pulses before and after the earthquake No. 3.

**Figure 9.**Dynamics of the probability density of distribution of the most powerful frequencies of geoacoustic pulses before and after the earthquake No. 4.

**Figure 10.**Dynamics of the probability density of distribution of the most powerful frequencies of geoacoustic pulses before and after the earthquake No. 5.

No. | Epicenter Coordinates | Date and Time, UT | Local Magnitude ${\mathit{M}}_{\mathit{L}}$ | Depth, km | Epicentral Distance R, km | Earthquake Group |
---|---|---|---|---|---|---|

1 | 52.84${}^{\circ}$ N, 159.62${}^{\circ}$ E | 2012.09.24 14:39:52.9 | 4.8 | 65 | 95 | 1 |

2 | 49.30${}^{\circ}$ N, 155.78${}^{\circ}$ E | 2012.11.01 06:57:19.7 | 5.5 | 76 | 444 | 2 |

3 | 51.53${}^{\circ}$ N, 160.08${}^{\circ}$ E | 2012.10.15 01:18:58.6 | 6.0 | 44 | 205 | 1 |

4 | 52.38${}^{\circ}$ N, 159.22${}^{\circ}$ E | 2015.12.16 08:20:15.5 | 5.0 | 72 | 95 | 1 |

5 | 52.40${}^{\circ}$ N, 159.08${}^{\circ}$ E | 2013.01.19 16:48:09.2 | 4.9 | 56 | 87 | 1 |

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**MDPI and ACS Style**

Lukovenkova, O.; Marapulets, Y.; Solodchuk, A.
Adaptive Approach to Time-Frequency Analysis of AE Signals of Rocks. *Sensors* **2022**, *22*, 9798.
https://doi.org/10.3390/s22249798

**AMA Style**

Lukovenkova O, Marapulets Y, Solodchuk A.
Adaptive Approach to Time-Frequency Analysis of AE Signals of Rocks. *Sensors*. 2022; 22(24):9798.
https://doi.org/10.3390/s22249798

**Chicago/Turabian Style**

Lukovenkova, Olga, Yuri Marapulets, and Alexandra Solodchuk.
2022. "Adaptive Approach to Time-Frequency Analysis of AE Signals of Rocks" *Sensors* 22, no. 24: 9798.
https://doi.org/10.3390/s22249798