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Mathematical model of non-linear volcanic tremors

非线性火山地震的数学模型

基本信息

批准号：
19540441
负责人：
TAKEO Minoru
金额：
$ 2.75万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2007
资助国家：
日本
起止时间：
2007 至 2009
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-19540441/
关键词：
地震現象微動噴火現象非線形時系列解析時系列

项目摘要

On September 1, 2004, a middle-scale eruption occurred at Mt.Asama. Before the eruption, we observed three long-period tremors with singular waveforms, which occurred at 4:25, 11:30, and 20:30 on June 24, 2004. The common characteristic of these tremors is that the tip of the waveform is sharp. This sharp-pointed waveform may suggest a non-linear dynamics of the source process. In addition, these singular tremors occurred at intervals of 7 to 9 hours on the same day, suggesting that the source process of these tremors is in common. In this paper, the dynamical structure and characteristics of these tremors are investigated by employing some reliable and robust techniques in the estimation of geometrical and dynamical parameters.Embedding by the method of time delays has become the standard procedure in non-linear dynamical system analysis of a single time series. The first step for the nonlinear analysis of a single time series is to reconstruct a topologically equivalent attractor to … More the original in a relatively low-dimensional delay-coordinate space. The key questions are how the minimum embedding dimension can be determined for reconstructing the original dynamics, and how we select the delay time. We employed some reliable and robust techniques in the estimation of optimum delay time and minimum embedding dimension. Concretely, we used higher-order correlations to select an optimum delay time (Albano et al., 1991). A practical method for determining minimum embedding dimension proposed by Cao (1997) was used in this paper. To verify this approach, we applied this method to Julian's non-linear tremor model, obtaining suitable embedding dimension and correlation dimension for Julian's system.To select the long-period component, we employ a FIR low-pass filter with a cut-off frequency of 0.4Hz. For the tremor occurred at 4:25, the optimum time lag of 0.24 sec and the minimum embedding dimension of 8 were obtained by employing these methods. We succeed in reconstructing an attractor of the tremors using these dimension and time lag. Then, we calculated a correlation integral curve of the reconstructed attractor, founding a scaling region over one decade with the correlation dimension of 2.04 plus minus 0.17. The correlation dimension converges a certain value as increasing the embedding dimension. This suggests that the time series is not random data and the correlation dimension is estimated correctly. We analyzed two other tremors and revealed the non-linear dynamics of long-period tremors using the embedding method of time delays and the surrogate data analysis, and made clear that there existed a deterministic non-linear dynamics in the tremor excitation, which could be modeled with the system dimension between 3 to 7 (prospective dimension 3 or 4).We also applied a non-linear prediction approach based on the theory of KM_2O-Langevin equation to evaluate a contribution from non-linearity of the system. In addition, we developed a new method for detecting and picking P- and S-wave signals automatically. Compared to methods currently in use, our method requires less assumption with properties of the data time series. We also developed a new approach for analysis of frequency structure of tremor based on the theory of KM_2O-Langevin equation. We applied the new algorithm to obtain a frequency structure form highly noisy data of deep low-frequency tremors occurred in western Shikoku, Japan, and reveal the characteristic frequency structure of deep low-frequency tremors with peaks lined up from 1 to 5Hz at intervals of 0.5Hz.In next step, we apply the embedding method of time delays to the long-period long-lasting earthquakes and estimate geometrical and dynamical non-linear parameters of them to constrain the dynamics in the excitation. Embedding by the method of time delays has become the standard procedure in non-linear dynamical system analysis of a single time series. The waveforms of the long-period long-lasting earthquakes were similar to each other, so we selected a typical event that occurred at 12:34 on June 12, 2004. We employed a FIR low-pass filter with a cut-off frequency of 1Hz to omit high frequency component. The optimum time lag of 0.24 sec and the minimum embedding dimension of 7 were obtained by employing these methods. We succeed in reconstructing the attractor of the long-period earthquakes, and got a correlation integral curve of the reconstructed attractor, founding a scaling region over one decade with the correlation dimension of 2.04 plus minus 0.11. This result indicates that the source process of long-period long-lasting earthquake could be modeled on a non-linear dynamics with a system dimension between 3 to 6, which is similar dimension range with the source process of long-period tremors. Another way of saying, the apparent waveform characteristics of long-period earthquakes and long-period tremors are quite different, however the both correlation dimensions calculated from the reconstructed attractors are almost same values.Modifying a hydraulic control valve model with the system dimension of 4, we succeeded to simulate a long-period oscillation resembling with the long-period earthquakes and with the long-period tremors based on a same mathematical model. These two long-period oscillations are sharply distinguished by a discharge coefficient of vent. The remaining problem is how to excite seismic waves from the simulated valve oscillations. Less

2004年9月1日，浅间山发生了一次中等规模的喷发。2004年6月24日4时25分、11时30分和20时30分，我们观测到三次奇异波形的长周期地震。这些震动的共同特征是波形的尖端是尖锐的。这种尖锐的波形可能表明源过程的非线性动力学。此外，这些单一地震在同一天每隔7 ~ 9小时发生一次，说明这些地震的震源过程是共同的。本文采用可靠的、鲁棒的几何参数和动力学参数估计技术，研究了这些地震的动力结构和特性。用时间延迟方法进行嵌入已经成为单时间序列非线性动力系统分析的标准方法。对单个时间序列进行非线性分析的第一步是在一个相对低维的延迟坐标空间中重构一个拓扑等价的吸引子，使其更接近原始的吸引子。关键问题是如何确定最小嵌入维数来重建原始动态，以及如何选择延迟时间。我们采用了一些可靠的鲁棒技术来估计最优延迟时间和最小嵌入维数。具体而言，我们使用高阶相关性来选择最佳延迟时间（Albano et al., 1991）。本文采用了Cao（1997）提出的确定最小嵌入维数的实用方法。为了验证该方法，我们将该方法应用于Julian的非线性震颤模型，得到了适合Julian系统的嵌入维数和相关维数。为了选择长周期分量，我们采用了截止频率为0.4Hz的FIR低通滤波器。对于发生在4点25分的地震，该方法得到的最佳滞后时间为0.24秒，最小嵌入维数为8。我们成功地利用这些维数和时间延迟重建了地震的吸引子。然后，我们计算了重构吸引子的相关积分曲线，建立了一个相关维数为2.04±0.17的十年尺度区域。随着嵌入维数的增加，相关维收敛到一定值。这说明时间序列不是随机数据，相关维数估计正确。我们分析了另外两个地震，利用时间延迟嵌入方法和替代数据分析揭示了长周期地震的非线性动力学，明确了地震激励存在确定性的非线性动力学，可以用系统维数3 ~ 7（预期维数3或4）来建模。我们还应用了基于km_20 - langevin方程理论的非线性预测方法来评估系统非线性的贡献。此外，我们还开发了一种自动检测和提取P波和s波信号的新方法。与目前使用的方法相比，我们的方法对数据时间序列性质的假设较少。本文还提出了一种基于km_20 - langevin方程理论分析地震频率结构的新方法。利用该算法从日本四国西部地区的高噪声深低频地震数据中获得了频率结构，揭示了以0.5Hz为间隔在1 ~ 5Hz范围内排列的深低频地震的特征频率结构。下一步，我们将时间延迟的嵌入方法应用于长周期持久地震，并估计其几何和动力学非线性参数来约束激励中的动力学。用时间延迟方法进行嵌入已经成为单时间序列非线性动力系统分析的标准方法。由于长周期持久地震的波形具有相似性，因此选取了2004年6月12日12时34分发生的一次典型地震。我们采用截止频率为1Hz的FIR低通滤波器来忽略高频成分。结果表明，该方法的最佳时滞为0.24秒，最小嵌入维数为7。我们成功地重建了长周期地震的吸引子，并得到了重建吸引子的相关积分曲线，建立了一个相关维数为2.04±0.11的十年尺度区域。这一结果表明，长周期持久地震的震源过程可以用一个系统维数在3 ~ 6之间的非线性动力学模型来模拟，其维数范围与长周期地震的震源过程相似。换句话说，长周期地震和长周期地震的表观波形特征有很大的不同，但从重建的吸引子计算出的相关维数几乎是相同的。对系统维数为4的液压控制阀模型进行修正，成功地模拟出了类似于长周期地震和基于同一数学模型的长周期地震的长周期振荡。这两种长周期振荡可以通过通风口的流量系数来明显区分。剩下的问题是如何从模拟的阀门振动中激发地震波。少