Statistics of Earthquakes

Earthquake statistics are facts recorded in the aftermath of seismic events. They are the concern of seismologists, geologists, engineers, government officials, insurers, and statisticians among others. Earthquakes provide special opportunties to learn about the makeup of the solid Earth. In the parlance of system identification, an earthquake is a pulse input at the event’s origin to the system (Earth) having as responses seismograms observed around the Earth. A seismogram is a recorded time series of the displacements, velocities, or accelerations experienced by a particle at a location of the Earth. Figure 1 presents an example. It is a part of a record of the Earth’s vertical motion as observed at Uppsala, Sweden, on April 20, 1989. It is highlighted here because of its use, together with a physical model and statistical methods, to learn about the surface composition of the Earth between Siberia and Upsalla (Bolt and Brillinger [12]). From seismograms recorded around Europe, using nonlinear regression analysis seismologists inferred that this event originated in the Sakha Republic of Russia. In the record one notes a variety of wiggles and fluctuations of varying amplitudes and periods. Seismologists attach physical significance to such features recording specific values such as arrival times of waves of differing types and routes through the Earth. The oscillations around the 30-second mark in the figure correspond to a Rayleigh wave train. Data concerning great earthquakes have been noted in China for more than two thousand years (Bolt [10]; Gu [25]). In particular there was a major collection of data immediately after the great Lisbon earthquake of 1755. These data are proving of high importance these days, as the papers in Mendes-Victor [47] show. Statistics and statisticians are involved because of the large amount of and many forms of data that become available following an earthquake as well as the related scientific and social questions arising. Statistical methods have played an important role in seismology for many years in part because of the pathbreaking efforts of Harold Jeffreys (see Bolt [9]). Concerning Jeffreys’ work, Hudson [30] has written: ‘‘The success of the Jeffreys–Bullen travel time tables was due in large part to Jeffreys’ consistent use of sound statistical methods. ’’ In particular, Jeffreys’ methods were robust and resistant, i.e., dealt with nongaussian distributions and outliers. Bolt [8] extended them to the linear regression case. Statistics enters for a variety of reasons. For example, the basic quantity of concern may be a probability model or a risk. Further, the data sets are often massive and of many types. Also there is a substantial inherent variability and measurement error. In response, these days seismologists and seismic engineers continually set down stochastic models. Consider, for example, the Next Generation of Attenuation (NGA) (Stewart [64]). Such models need to be fitted, assessed, and revised. Inverse problems with the basic parameters defined indirectly need to be solved (O’Sullivan [53]; Stark [63]). Experiments need to be designed. In many cases researchers employ simulations and massive databases of such have been developed (see Olsen and Ely [52]). It can be noted that new statistical techniques often find immediate application in seismology particularly and in geophysics generally. In parallel, problems arising in seismology and earthquake engineering have led to the development of new statistical techniques. Seismology underwent the ‘‘digital revolution’’ in the nineteen-fifties and continually poses problems exceeding the capabilities of the day’s computers. Its participants have turned up a variety of empirical laws (Kanamori [37]). These prove useful for extrapolation to situations with few data (e.g., Huyse [32], Zhuang [78], and Amorese [3]). Physical theories find important application (Aki and Richards [1]). The subject matter developed leads to hazard estimation (Wesnowski [76]); improved seismic (Naeim [48]; Mendes-Victor [47]); earthquake prediction (Zechar [77]; Lomnitz [42]; Harte and Vere-Jones [28]; Luen and Stark [43]); determination