Yosihiko Ogata

Space-Time Point-Process Models for Earthquake Occurrences

Several space-time statistical models are constructed based on both classical empirical studies of clustering and some more speculative hypotheses. Then we discuss the discrimination between models incorporating contrasting assumptions concerning the form of the space-time clusters. We also examine further practical extensions of the model to situations where the background seismicity is spatially non-homogeneous, and the clusters are non-isotropic. The goodness-of-fit of the models, as measured by AIC values, is discussed for two high quality data sets, in different tectonic regions. AIC also allows the details of the clustering structure in space to be clarified. A simulation algorithm for the models is provided, and used to confirm the numerical accuracy of the likelihood calculations. The simulated data sets show the similar spatial distributions to the real ones, but differ from them in some features of space-time clustering. These differences may provide useful indicators of directions for further study.

On Lewis' simulation method for point processes

A simple and efficient method of simulation is discussed for point processes that are specified by their conditional intensities. The method is based on the thinning algorithm which was introduced recently by Lewis and Shedler for the simulation of nonhomogeneous Poisson processes. Algorithms are given for past dependent point processes containing multivariate processes. The simulations are performed for some parametric conditional intensity functions, and the accuracy of the simulated data is demonstrated by the likelihood ratio test and the minimum Akaike information criterion (AIC) procedure.

Stochastic Declustering of Space-Time Earthquake Occurrences

This article is concerned with objective estimation of the spatial intensity function of the background earthquake occurrences from an earthquake catalog that includes numerous clustered events in space and time, and also with an algorithm for producing declustered catalogs from the original catalog. A space-time branching process model (the ETAS model) is used for describing how each event generates offspring events. It is shown that the background intensity function can be evaluated if the total spatial seismicity intensity and the branching structure can be estimated. In fact, the whole space-time process is split into two subprocesses, the background events and the clustered events. The proposed algorithm combines a parametric maximum likelihood estimate for the clustering structures using the space-time ETAS model and a nonparametric estimate of the background seismicity that we call a variable weighted kernel estimate. To demonstrate the present methods, we estimate the background seismic activities in the central region of New Zealand and in the central and western regions of Japan, then use these estimates to produce catalogs of background events.

Whole mantle P-wave travel time tomography

Abstract A method of tomographic inversion to obtain three-dimensional velocity perturbations in the Earths whole mantle has been developed, and applied to more than two million P-wave arrival time data reported by International Seismology Center (ISC). The model is parameterized with 32,768 blocks; the divisions in latitude, longitude, and radius are 32, 64, and 16, respectively. Horizontal cell size is 5.6° × 5.6°. The layer thicknesses vary with depth; 29 km just below the surface and 334 km just above the core-mantle boundary. Starting from a spherically symmetric Earth model, we obtained a three-dimensional model using the following iterative procedures. First, we relocated all the events; second, we backprojected the residual into the whole mantle; third, we refined the spherically symmetric Earth model. The solutions have been converged in five iterations. We adopted the following techniques in the backprojection procedure. The first order smoothness was introduced as a damping, which makes the solution independent of the starting model and its apparent fluctuation minimal. The basic equations for delay times and smoothness are solved using the conjugate gradient method, an iterative method which guarantees the convergence of solution into the exact least squares solution. The weight on the smoothness, i.e., the damping factor, was objectively determined by a simplified cross validation technique. The final solution was obtained as an average of the ten solutions, each of which was derived from one tenth of the total data set. The reliability of the solution is examined in two ways: (1) mapping the resolution given by the reconstruction of checkerboard patterns, and (2) mapping the variance given from the gaussian noise input. This is the first result which delineates the whole mantle structure with its reliability mapping. The longest wavelength anomalies of the lower mantle are similar to those of previous studies by Dziewonski (1984) and Hager and Clayton (1989). However, the shorter wavelength patterns have many discrepancies. The fast anomalies of the uppermost mantle beneath the shield and the slow anomalies beneath the active tectonic regions are clearly seen. We also revealed the l = 2 pattern at the transition zone reported by the previous surface wave and free oscillation studies. The most striking feature is that of the fast anomalies lying along the extension of the large subducting plate up to depths of about 1000 km or more. The resolving powers in these regions are generally good.

Detection of precursory relative quiescence before great earthquakes through a statistical model

Relative quiescence is defined as a significant decrease of earthquakes compared with the occurrence rate expected from a point-process model for ordinary seismic activity. This definition is linked with the idea that aftershocks, as a useful signal for the investigation of seismicity, should not be removed from original occurrence data. The Epidemic Type Aftershock-Sequences (ETAS) model is proposed to identify patterns in an earthquake sequence in an area, using occurrence time and magnitude data. Even in an active stage of seismicity, a decrease from the expected level can take place, and the significance and size of such a quiescence can be graphically shown using time-changed occurrence data which are transformed based on the estimated ETAS model. This procedure permits detection of a clear, relatively quiet stage before great earthquakes in Japan and elsewhere in the world. Such relative quiescence lasts a number of years prior to the occurrence of great earthquakes, which occasionally take place in the recovering stage. No significant relative quiescences were seen in the current seismicity of a few wide regions, in and around Japan, including several source regions of expected great earthquakes.

Statistical relations between the parameters of aftershocks in time, space, and magnitude

The correlation between statistical parameters of seismicity, such as the b value of the Gutenberg and Richter [1954] relation, the p value of the modified Omori formula [Utsu, 1961], the p and α value of the Epidemic Type Aftershock Sequence (ETAS) model [Ogata, 1992], and the fractal dimension D of the hypocenter distribution, is analyzed for 34 aftershock sequences in Japan from 1971 to 1995. All the parameters are estimated using maximum likelihood methods along with their error assessments. For the majority of the aftershock sequences, the ETAS model fits statistically better than the modified Omori formula, which suggests existence of clusters within the aftershock sequence. Most of the scatterplots between the estimates of the seismicity parameters in time, space, and magnitude distributions are clearly seen to be either positively or negatively correlated. The contrasting correlation patterns are revealed between the estimated parameters for the intraplate and interplate earthquakes, except for the two pairs (b,D) and (α,p) in which similar correlation patterns are found. We focus our attention on these patterns as a source of interesting contrasts between the two earthquake groups. In particular, the significant dependence of these parameters on the depth appears to be a key to understanding the correlation pattern for interplate aftershocks, while a different interpretation is made for intraplate aftershocks because no significant dependence on depth is observed.

Statistical model for standard seismicity and detection of anomalies by residual analysis

Abstract A statistical point-process model is derived to describe the standard activity of earthquake occurrences by assuming that general seismicity is given by the superposition of aftershock sequences. The parameters are estimated ty the maximum likelihood method. Using the estimated model, the “residual point process” of the data is defined and used to find the anomalies which are included in the data set but not captured in the considered model for the standard seismicity. For instance, seismic quiescences can be measured quantitatively by using the residual process. Some examples are provided to illustrate such analyses. Furthermore, a time series of the magnitudes on the residual point process is considered, to investigate its dependence either on the time or on the history of the seismicity. By assuming the exponential distribution at each time and modelling of the b- value , we can examine such dependences and estimate them. Two practical examples are shown.

Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure

SummaryA homogeneous spatial point pattern is regarded as one of thermal equilibrium configurations whose points interact on each other through a certain pairwise potential. Parameterizing the potential function, the likelihood is then defined by the Gibbs canonical ensemble. A Monte Carlo simulation method is reviewed to obtain equilibrium point patterns which correspond to a given potential function. An approximate log likelihood function for gas-like patterns is derived in order to compute the maximum likelihood estimates efficiently. Some parametric potential functions are suggested, and the Akaike Information Criterion is used for model selection. The feasibility of our procedure is demonstrated by some computer experiments. Using the procedure, some real data are investigated.

A study on the background and clustering seismicity in the Taiwan region by using point process models

[1] This paper investigates the shallow seismicity occurring in the Taiwan region during the 20th century using a stochastic declustering method that has been developed on the basis of the theory of the epidemic-type aftershock sequence model. It provides a probability based tool to objectively separate the space-time occurrences of earthquakes into a background and a clustering component. On the basis of the background and clustering seismicity rates, we discuss the correlation between the distribution of the cluster ratio and the regional seismotectonic structures. Specifically, we find that the areas of the highest clustering ratio correspond to the major strike-slip fault traces in and around Taiwan. Additionally, in the Taiwan inland region, during the period 1960-1990, the outputs for the stochastically declustered catalogue show a clear quiescence in background seismicity preceding the recovery of activation and the occurrences of the 1999 Chi-Chi earthquake of M L 7.3, while the other active regions show stationary background activity. This could be interpreted as an effect of the aseismic slip in the Chi-Chi rupture fault, whereby the inland region around the Chi-Chi source becomes a stress shadow.

On Parameter Estimation for Pairwise Interaction Point Processes

Summary Pairwise interaction point processes form a useful class of models for spatial point patterns, especially patterns for which the spatial distribution of points is more regular than for a homogeneous planar Poisson process. Several authors have proposed methods for estimating the parameters of a pairwise interaction point process. However, there appears to be no general theory which provides grounds for preferring a particular method, nor have any extensive empirical comparisons been published. In this paper, we review three general methods of estimation which have been proposed in the literature and present the results of a comparative simulation study of the three methods.