  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.
