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The detection of earthquakes is a fundamental prerequisite for seismology and contributes to various research areas, such as forecasting earthquakes and understanding the crust/mantle structure. Recent advances in machine learning technologies have enabled the automatic detection of earthquakes from waveform data. In particular, various state-of-the-art deep-learning methods have been applied to this endeavour. In this study, we proposed and tested a novel phase detection method employing deep learning, which is based on a standard convolutional neural network in a new framework. The novelty of the proposed method is its separate explicit learning strategy for global and local representations of waveforms, which enhances its robustness and flexibility. Prior to modelling the proposed method, we identified local representations of the waveform by the multiple clustering of waveforms, in which the data points were optimally partitioned. Based on this result, we considered a global representation and two local representations of the waveform. Subsequently, different phase detection models were trained for each global and local representation. For a new waveform, the overall phase probability was evaluated as a product of the phase probabilities of each model. This additional information on local representations makes the proposed method robust to noise, which is demonstrated by its application to the test data. Furthermore, an application to seismic swarm data demonstrated the robust performance of the proposed method compared with those of other deep learning methods. Finally, in an application to low-frequency earthquakes, we demonstrated the flexibility of the proposed method, which is readily adaptable for the detection of low-frequency earthquakes by retraining only a local model.

A seismic wavefield reconstruction framework based on compressed sensing using the data-driven reduced-order model (ROM) is proposed and its characteristics are investigated through numerical experiments. The data-driven ROM is generated from the dataset of the wavefield using the singular value decomposition. The spatially continuous seismic wavefield is reconstructed from the sparse and discrete observation and the data-driven ROM. The observation sites used for reconstruction are effectively selected by the sensor optimization method for linear inverse problems based on a greedy algorithm. The proposed framework was applied to simulation data of theoretical waveform with the subsurface structure of the horizontally-stratified three layers. The validity of the proposed method was confirmed by the reconstruction based on the noise-free observation. Since the ROM of the wavefield is used as prior information, the reconstruction error is reduced to an approximately lower error bound of the present framework, even though the number of sensors used for reconstruction is limited and randomly selected. In addition, the reconstruction error obtained by the proposed framework is much smaller than that obtained by the Gaussian process regression. For the numerical experiment with noise-contaminated observation, the reconstructed wavefield is degraded due to the observation noise, but the reconstruction error obtained by the present framework with all available observation sites is close to a lower error bound, even though the reconstructed wavefield using the Gaussian process regression is fully collapsed. Although the reconstruction error is larger than that obtained using all observation sites, the number of observation sites used for reconstruction can be reduced while minimizing the deterioration and scatter of the reconstructed data by combining it with the sensor optimization method.

The ``big'' seismic data not only acquired by seismometers but also acquired by vibrometers installed in buildings and infrastructure and accelerometers installed in smartphones will be certainly utilized for seismic research in the near future. Since it is impractical to utilize all the seismic big data in terms of the computational cost, methods which can select observation sites depending on the purpose are indispensable. We propose an observation site selection method for the accurate reconstruction of the seismic wavefield by process-driven approaches. The proposed method selects observation sites suitable for accurately estimating physical model parameters such as subsurface structures and source information to be input into a numerical simulation of the seismic wavefield. The seismic wavefield is reconstructed by the numerical simulation using the parameters estimated based on the observed signals at only observation sites selected by the proposed method. The observation site selection in the proposed method is based on the sensitivity of each observation site candidate to the physical model parameters; the matrix corresponding to the sensitivity is constructed by approximately calculating the derivatives based on the simulations, and then, observation sites are selected by evaluating the quantity of the sensitivity matrix based on the D-optimality criterion proposed in the optimal design of experiments. In the present study, physical knowledge on the sensitivity to the parameters such as seismic velocity, layer thickness, and hypocenter location was obtained by investigating the characteristics of the sensitivity matrix. Furthermore, the effectiveness of the proposed method was shown by verifying the accuracy of seismic wavefield reconstruction using the observation sites selected by the proposed method., Comment: Accepted manuscript for publication in Geophysical Journal International

Long-term slow-slip events (LSSEs) usually occur on the deep, shallow parts of subducting plates and have substantial relation with adjacent megathrust fault motion. Conventional techniques of quantifying slow earthquake frictional features show that these features may be indicative of predictive seismic motion; however, quantifying high-accuracy uncertainty of the frictional fields has not yet been achieved. We therefore propose a method of uncertainty quantification for spatially inhomogeneous frictional features from slip motion on an LSSE fault--megathrust fault complex in southwestern Japan. By combining a fault motion model that mimics slow-slip motion and a variational data assimilation (DA) technique using a second-order adjoint method, we have succeeded in quantifying the spatial distribution of the uncertainty of the frictional features. Further, evaluation of the spatial distribution in high-resolution reveals the correlation between the dynamics of the slow-slip motion and the important components of the frictional features, which is valuable information for observational DA design. Findings from this study are expected to advance the theoretical foundation of applied seismic motion prediction techniques using slow-slip frictional features as stress meters for megaquakes, as well as improve understanding of the relationship between the slow-slip motion and frictional parameters of a fault.

We propose a phase prediction method for the pattern formation in the uniaxial two-dimensional kinetic Ising model with the dipole-dipole interactions under the time-dependent Ginzburg-Landau dynamics. Taking the effects of the material thickness into account by assuming the uniformness along the magnetization axis, the model corresponds to thin magnetic materials with long-range repulsive interactions. We propose a new theoretical basis to understand the effects of the material parameters on the formation of the magnetic domain patterns in terms of the equation of balance governing the balance between the linear- and nonlinear forces in the equilibrium state. Based on this theoretical basis, we propose a new method to predict the phase in the equilibrium state reached after the time-evolution under the dynamics with a given set of parameters, by approximating the third-order term using the restricted phase-space approximation [R. Anzaki, K. Fukushima, Y. Hidaka, and T. Oka, Ann. Phys. 353, 107 (2015)] for the $\phi^4$-models. Although the proposed method does not have the perfect concordance with the actual numerical results, it has no arbitrary parameters and functions to tune the prediction. In other words, it is a method with no a priori assumptions on domain patterns.

Uncovering the distribution of magnitudes and arrival times of aftershocks is a key to comprehend the characteristics of the sequence of earthquakes, which enables us to predict seismic activities and hazard assessments. However, identifying the number of aftershocks immediately after the main shock is practically difficult due to contaminations of arriving seismic waves. To overcome the difficulty, we construct a likelihood based on the detected data incorporating a detection function to which the Gaussian process regression (GPR) is applied. The GPR is capable of estimating not only the parameters of the distribution of aftershocks together with the detection function but also credible intervals for both of the parameters and the detection function. A property that distributions of both the Gaussian process and aftershocks are exponential functions leads to an efficient Bayesian computational algorithm to estimate the hyperparameters. After the validations through numerical tests, the proposed method is retrospectively applied to the catalog data related to the 2004 Chuetsu earthquake towards early forecasting of the aftershocks. The result shows that the proposed method stably estimates the parameters of the distribution simultaneously their credible intervals even within three hours after the main shock.

Many signals on Cartesian product graphs appear in the real world, such as digital images, sensor observation time series, and movie ratings on Netflix. These signals are "multi-dimensional" and have directional characteristics along each factor graph. However, the existing graph Fourier transform does not distinguish these directions, and assigns 1-D spectra to signals on product graphs. Further, these spectra are often multi-valued at some frequencies. Our main result is a multi-dimensional graph Fourier transform that solves such problems associated with the conventional GFT. Using algebraic properties of Cartesian products, the proposed transform rearranges 1-D spectra obtained by the conventional GFT into the multi-dimensional frequency domain, of which each dimension represents a directional frequency along each factor graph. Thus, the multi-dimensional graph Fourier transform enables directional frequency analysis, in addition to frequency analysis with the conventional GFT. Moreover, this rearrangement resolves the multi-valuedness of spectra in some cases. The multi-dimensional graph Fourier transform is a foundation of novel filterings and stationarities that utilize dimensional information of graph signals, which are also discussed in this study. The proposed methods are applicable to a wide variety of data that can be regarded as signals on Cartesian product graphs. This study also notes that multivariate graph signals can be regarded as 2-D univariate graph signals. This correspondence provides natural definitions of the multivariate graph Fourier transform and the multivariate stationarity based on their 2-D univariate versions.

Data assimilation (DA) is a fundamental computational technique that integrates numerical simulation models and observation data on the basis of Bayesian statistics. Originally developed for meteorology, especially weather forecasting, DA is now an accepted technique in various scientific fields. One key issue that remains controversial is the implementation of DA in massive simulation models under limited computation time and resources. In this paper, we propose an adjoint-based DA method for massive autonomous models that produces optimum estimates and their uncertainties within practical computation time and resource constraints. The uncertainties are given as several diagonal components of an inverse Hessian matrix, which is the covariance matrix of a normal distribution that approximates the target posterior probability density function in the neighborhood of the optimum. Conventional algorithms for deriving the inverse Hessian matrix require $O(CN^2+N^3)$ computations and $O(N^2)$ memory, where $N$ is the number of degrees of freedom of a given autonomous system and $C$ is the number of computations needed to simulate time series of suitable length. The proposed method using a second-order adjoint method allows us to directly evaluate the diagonal components of the inverse Hessian matrix without computing all of its components. This drastically reduces the number of computations to $O(C)$ and the amount of memory to $O(N)$ for each diagonal component. The proposed method is validated through numerical tests using a massive two-dimensional Kobayashi's phase-field model. We confirm that the proposed method correctly reproduces the parameter and initial state assumed in advance, and successfully evaluates the uncertainty of the parameter. Such information regarding uncertainty is valuable, as it can be used to optimize the design of experiments.