Hiromichi Nagao

Tomoki Tokuda, Hiromichi Nagao

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.

Takashi Kurokawa, Taihei Oki, Hiromichi Nagao

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.