Nonlinear analysis of the frequency-magnitude relationship in the western circum-pacific region

It has long been realized that the linear Gutenberg-Richter model arduously describes the frequency-magnitude relationship for the magnitude span ranging from small to large earthquakes because of the breakdown of the self-similarity rule due to the changing scaling of the magnitude. Three different segments should be observed from small (usually M<3.0), through moderate (M<M-c, where M-c is the frequency-magnitude turning point caused by the seismogenic thickness), to large earthquakes (Mgreater than or equal toM(c)). We will only concentrate on the moderate and large earthquakes due to their importance. The breakdown of the self-similarity rule from moderate to large earthquakes occurs where the earthquake is big enough to cut through the entire seismogenic layer. A nonlinear 'hyperbolic' model, which fits two linear relations smoothly, log N=a(1)+a(2)(M-a(4))+a(3)[(M-a(4))(2)+a(5)](1/2) is studied in the present paper, where N is the cumulative number of earthquakes with magnitudes larger than or equal to M; a(1) to a(5) are constants to be calculated. The G-R linear relation is actually a special case of the present nonlinear model, i.e., a(3) or a(5) equal to zero. The nonlinear form, with the support of a reasonable physical mechanism, can generally give a better fitting with comparatively minor errors for complete data sets, especially for the areas where large earthquakes are numerous. In order to demonstrate its superiority to the linear G-R relation, thirteen seismogenic zones are examined around the western part of the Circum-Pacific region and western part of China and it is found that the fitting errors from this nonlinear model are, as expected, generally much smaller than those for G-R. Furthermore, the parameter a(4) is believed to relate with the saturated magnitude M(c,)which to some extent reflects the mean thickness of the seismogenic layer.