Statistical models are introduced for the simultaneous estimation of the changes of the b value and detection rate of earthquakes in a catalogue which develops in time or varies in space. The three characteristic parameters, including b, for the magnitude frequency distribution of detected earthquakes, are represented by respective three B-spline functions of time or location in a space. An objective Bayesian method is adopted for the optimal estimate of such functions with many unknown coefficients. The present procedure is applied to earthquake catalogues for Japanese seismic activity.

Earthquake intensities are modelled as a function of previous activity whose specific form is based on established empirical laws in seismology, but whose parameter values can vary from place to place. This model is used for characterizing regional features of seismic activities in and around Japan, and also for exploring regions where the actual seismicity rate systematically deviates from that of the modelled rate. Copyright 2003 Royal Statistical Society.

Marked point patterns, intensity, distributions of marks, B-splines, smoothing, penalized likelihood, objective Bayesian method, magnitude frequency, b-value,

S U M M A R YA method is developed for estimation and interpolation of 6-values in space. A 3-D spline function is considered for the logarithm of the 6-value at each location in the space. Since many parameters for the spline coefficients are required to obtain a sensible estimate of the spatial variation of 6-values, we consider the penalized log-likelihood with the standard roughness penalties for the spline function. Further the error bands of the b-value estimation at each location can be calculated. Using the current method, the spatial distribution of 6-values beneath the Kanto District down to the depth of 100km is determined based on hypocentral data of microearthquakes from the Kanto-Tokai Observational Network of the National Research Center for Disaster Prevention. The stability of the estimated pattern is checked by comparing with the results using alternative cut-off magnitudes. This is further ensured by comparison with the result obtained by an alternative model using equally divided blocks. On the whole, the vertical change in 6-value is greater than the horizontal one. It is high in the crust of the Eurasian plate, especially above the upper boundary of the subducting Pacific plate and in the northwest part, or the volcanic area, in the Kanto District. A steep decrease of the 6-values is seen to take place in perpendicular direction to the subducting Pacific plate boundary. Also, a similar change is seen in the boundary between the Eurasian and Phillippine Sea plates, especially beneath the southern part of the Kanto Plain. The 6-value is low in the upper boundary of the subducting plates, but high in the lower plane of the double seismic zone in the Pacific plate. It appears that, even within a narrow area of aftershocks, the 6-value can change significantly. It is also found that the variation of the 6-value estimate is in good agreement with the structure of seismic wave fractional velocity perturbations. The regions of high and low 6-values correspond, respectively, to the lower and higher parts of the P-wave velocity. The similar relationship is seen with the spatial structure of the seismic wave attenuation.

[1] It is well known that the detection rate of aftershocks is extremely low during the period immediately after a large earthquake due to the contamination of arriving seismic waves. This has led to considerable difficulties in obtaining estimates of the empirical laws of aftershock decay and magnitude frequency immediately after main shocks. This paper presents an estimation method for predicting the underlying occurrence rate of aftershocks of any magnitude range, based on a magnitude frequency model that combines the Gutenberg-Richter law with the detection rate function. This procedure enables real-time probability forecasting of aftershocks immediately after the mainshock, when the majority of large aftershocks are likely to occur.