Nonlinear Analysis of the Frequency-magnitude Relationship in the Western Circum-Pacific Region

Abstract

— 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 (M ≥ M _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, 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., a₃ or a₅ 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 a₄ is believed to relate with the saturated magnitude M _c, which to some extent reflects the mean thickness of the seismogenic layer.