Stochastic models and forecast for recurrent earthquakes 繰り返し地震の確率論的モデルと予測手
Stochastic models and forecast for recurrent earthquakes
This thesis deals with recurrent earthquakes. Many earthquakes occur repeatedly at the same hypocenter quasi-periodically to some extent. So, the renewal process that assumes quasi-periodicity fits better than Poisson process in many cases. Though renewal process is one of the most popular models for earthquake forecast, there are some problems in forecasting catastrophic large earthquakes. This thesis gives some solutions for the following problems: 1. scarcity of occurrence data, 2. uncertainty of occurrence data and 3. change of recurrence structure in space and time. They have been difficult problems to apply renewal processes to earthquake forecast. Thus, this thesis deals with these problems and gives some solutions to overcome them. The first problem, the scarcity of occurrence data, is prominent for large earthquakes. Catastrophic earthquakes recur by hundreds or thousands of years and often have one or a few records in active faults. It causes large error in parameter estimation and earthquake forecast. So we propose a Bayesian model to forecast the next earthquake. When only limited records of characteristic earthquakes on a fault are available, relevant prior distributions for renewal model parameters are essential to computing unbiased, stable time‐dependent earthquake probabilities. From various prior models, we select the common prior distribution that has the smallest value of the Akaike’s Bayesian Information Criterion (ABIC) (Akaike, 1980). We also use geological information, such as single earthquake displacements (U) and deformation rate (V) to calculate mean recurrence time as T = U/V in addition to recurrence intervals obtained directly from historical records and paleoseismic investigations (Rhoades et al., 1994). By comparing the goodness of fit to the historical record and simulated data, we show that the proposed predictor provides more stable performance than plug-in predictors, such as maximum likelihood estimates and the predictor currently adopted by the ERC. The second problem, uncertainty of occurrence data, is seen in paleoearthquakes. Since paleoseismic investigation specifies the trace of seismic activities in stratum and infers its occurrence date from radiocarbon age of the surrounding deposits, only the upper and lower limits are specified for the occurrence date. When the estimated ranges for occurrence date are so wide, they affect probability forecast critically. Additionally, it is often the case that it is hard to judge whether earthquake occurred or not in a layer accumulated in some period of age. Even if we could specify the trace of earthquakes, there is a case that it is hard to specify how many earthquakes had occurred. In these cases, the dataset have uncertainty of occurrence itself as well as occurrence date and we have to consider them to analyze the data. To use all information from historical accounts and paleoseismic investigations, these uncertainties should be incorporated into stochastic model. Thus, we consider a likelihood function of data sets with various kinds of uncertainties for previous Bayesian model and forecast next earthquakes by the Bayesian predictive distribution. We evaluate the probability forecast with the BPT renewal model where the ERC applies not renewal process but Poisson process due to the large uncertainty of the occurrence date and showed that our model can give an additional information for the uncertain occurrence date. Although there remains large uncertainty in our evaluation due to the scarcity of the data, it gives us the extent of the evaluation changed by additional data or information. The last problem, change of recurrence structure, is seen for earthquakes repeating in shorter period. When we analyze the sequence of earthquakes repeating in shorter period, we will confront with the problem that the inter-event times have some trends which develop by time. It is caused by the change in geodetic deformation rate or larger earthquakes, and so repeating earthquakes at nearby hypocenters show similar trends on the inter-event times. Therefore, we introduce the space-time renewal model for repeating earthquakes. It incorporates the space-time structure represented by 3-dimensional cubic B-spline functions into the renewal model. Our model has enormous coefficient parameters for B-spline bases and they are estimated by maximizing the integrated likelihood in Bayesian framework. In order to avoid the overfitting, we prepare the smoothness prior distribution which penalizes for the differential of B-spline functions and select its hyper-parameters by their ABIC values. We analyze the repeating earthquakes in Parkfield segment of San Andreas Fault specified by Nadeau et al. (1995) and estimate the change of stressing rate in space and time before and after 2004 Parkfield earthquake. The results show the trend of inter-event times reflects the nearby main earthquakes and aseismic creepment in this segment.