Mathematical model of non-linear volcanic tremors

2004年9月1日，浅间山发生了一次中等规模的喷发。在喷发前，我们观察到三个长周期的震颤与奇异波形，这发生在2004年6月24日4时25分，11时30分，20时30分。这些震颤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，这些奇异地震在同一天每隔7至9小时发生一次，表明这些地震的震源过程是共同的。本文采用一些可靠的、鲁棒的几何参数和动力参数估计方法，研究了这些地震的动力结构和特性，时滞嵌入方法已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构造一个拓扑等价的吸引子， ...更多信息 在一个相对低维的延迟坐标空间中的原始。关键问题是如何确定重构原动力学的最小嵌入维数，以及如何选择延迟时间。在最佳延迟时间和最小嵌入维数的估计中，我们采用了一些可靠的和鲁棒的技术。具体地，我们使用高阶相关性来选择最佳延迟时间（Albano等人，1991年）。本文采用了Cao（1997）提出的一种确定最小嵌入维数的实用方法。为了验证该方法的有效性，我们将该方法应用于Julian的非线性震颤模型，得到了适合Julian系统的嵌入维数和关联维数，并采用了一个截止频率为0.4Hz的FIR低通滤波器来选择长周期分量。对于4：25发生的地震，最佳时滞为0.24秒，最小嵌入维数为8。我们成功地重建吸引子的震颤使用这些尺寸和时间滞后。然后，我们计算了重建的吸引子的相关积分曲线，发现了一个十年的相关维数为2.04 ± 0.17的标度区域。随着嵌入维数的增加，关联维数收敛于某一值。这表明时间序列不是随机数据，相关维数估计正确。通过对另外两次地震的分析，利用时滞嵌入法和替代数据分析方法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非线性动力学，这可以用3到7之间的系统维度来建模我们还应用了基于KM_2 O-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，我们开发了一种新的方法，自动检测和拾取P波和S波信号。与目前使用的方法相比，我们的方法需要更少的假设与数据时间序列的属性。基于KM_2 O-Langevin方程理论，提出了一种分析地震动频率结构的新方法。应用该算法对日本四国西部地区深低频地震的高噪声数据进行了频率结构分析，揭示了深低频地震的特征频率结构，其峰值在1 ~ 5 Hz之间，间隔为0.5Hz。将时滞嵌入方法应用于长周期长持续地震，估计了地震的几何和动力非线性，它们的线性参数来约束激励中的动态。采用时间延迟方法进行嵌入已成为单时间序列非线性动力系统分析的标准程序。长周期长持续地震的波形相似，因此我们选择了2004年6月12日12时34分发生的一个典型地震。我们采用了一个FIR低通滤波器，截止频率为1Hz，以忽略高频分量。采用这些方法得到的最佳时间延迟为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，建立了一个十年以上的标度区，其相关维数为2.04 ± 0.11。这一结果表明，长周期长持续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，这与长周期地震的震源过程维数范围相近。换句话说，长周期地震和长周期地震的波形特征是不同的，但由重构的吸引子计算的两个关联维数几乎是相同的。我们成功地模拟了一个类似于长周期地震的长周期振荡和基于同一数学模型的长周期震颤。这两个长周期振荡明显区别的排放系数的通风。剩下的问题是如何从模拟的阀门振荡中激发地震波。少

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