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.

S U M M A R YWhen earthquakes occur in a cluster, the probability that they will be foreshocks of a forthcoming significantly larger earthquake appears dependent on the magnitude differences, origin-time spans, and distances between several of the earliest events in a cluster. The earthquake catalogue of the Japan Meteorological Agency (JMA;1926-91, MJ 2 4) is decomposed into numbers of clusters in time and space to compare statistical features of foreshocks with those of swarms and main shockaftershock sequences. Since the number of foreshocks in individual earthquake clusters is usually small, two types of data stacking are considered. First, we consider the spatial and temporal distribution of events relative to the location and time of the coming main shock. For the foreshocks, a relative doughnut-shaped pattern which converges in time to the main shock's epicentre is seen, while this pattern is weak for swarms. Next, we consider the temporal, spatial and magnitude distributions of events relative to the time, location and magnitude of earlier events in every sequence. Then, a statistical discrimination of foreshocks from earthquakes of other types of clusters is explored.The search for significant trends of relative frequency of foreshocks in comparison with other cluster types revealed the following features of foreshocks: (1) their relative frequency is high when the inter-event time span is less than several days, but the maximum is attained at about several hours span; (2) their relative frequency is high at short distances between epicentres, especially within 10 km, and then decreases as the distance gets larger; and (3) their relative frequency increases as the magnitude increases among the earliest shocks in a cluster. The stability of the results is confirmed by finding similar trends using different cluster identification methods and different threshold magnitudes. Further, microearthquakes in the Japan University Network data file (1983-87; M 2 2) and moderate-to-large earthquakes in the International Seismological Centre world catalogue (ISC; 1979-90, Mh 2 5 ) are examined to see some effects of the magnitude-thresholds of these catalogues. Similar features to those obtained in the JMA catalogue are also found for these catalogues.

SummaryThe computational aspect of the fitting of a parametric model for the analysis of the influence of an input to a point process output is discussed. The feasibility of the procedure is demonstrated by an artificial example. Its practical utility is illustrated by applying it to the analysis of the causal relation between two earthquake series data from certain seismic regions of Japan. l. IntroductionConsider a point process defined by the intensity process (i.I) ~(t)= F + f'og(t-s)dN~+ fto h(t-s)dXs ,where {N~} denotes the point process and {X~} the input process which may be either a point process or a cumulative process Xt= fto x(s)dsof a stochastic process x(t). Given bivariate data {N~,X,; O<_t

S U M M A R YWhen earthquake activity begins, it may be a foreshock sequence to a larger earthquake, a swarm, or a simple main-shock-aftershock sequence. This paper is concerned with the conditional probability that it will be foreshock activity of a later larger earthquake, depending on the occurrence pattern of some early events in the sequence. The earthquake catalogue of the Japan Meteorological Agency , M , 2 4 ) is decomposed into a large number of clusters in time and space in order to compare statistical features of foreshocks with those of swarms and aftershocks. Using such a data set, Ogata, Utsu & Katsura (1995) revealed some discriminating features of foreshocks relative to the other types of clusters, for example the events' closer proximity in time and space, and a tendency towards chronologically increasing magnitudes, which encouraged us to construct models which forecast the probability of the earthquakes being foreshocks. Specifically, the probability is a function of the history of magnitude differences, spans between origin times and distances between epicentres within a cluster. For purposes of illustration, the models were fitted to the early part of the data and the validity of the forecasting procedure was checked on data from the later period (1976)(1977)(1978)(1979)(1980)(1981)(1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993). Two procedures for evaluating the performance of the probability forecast are suggested. Furthermore, for the case where only a single event is available (i.e. either it is the first event in a cluster or an isolated event), we also forecast the probability of the event being a foreshock as a function of its geographic location. Then, the validity of the forecast is demonstrated in a similar manner. Finally, making use of the multi-element prediction formula, we show that the forecasting performance is enhanced by the joint use of the information in the location of the first event, and that in the subsequent interevent history in the cluster.