Estimation of Maximum Magnitude (c-value) and its Certainty for Modified Gutenberg-Richter Formulas, Based on Historical and Instrumental Japanese Intraplate Earthquake Catalogs

A-, b-, and c-values for the original Gutenberg-Richter formula (GR) and modified GR formulas (Utsu, 1978) were estimated using a dataset of combined historical (1595-1925 A.D.) and instrumental (1926-2000) Japanese earthquake data for 18 intraplate seismo-tectonic provinces depicted on a new tectonic map of Japan (Kakimi et al., 2002). The theoretical relationships between the b-values of the original and modified GR formulas, and the certainty of b- and c-values, were evaluated with respect to the dataset. The GR formula generally used for earthquake magnitude and frequency relationships demonstrates that earthquake frequency in each magnitude class is about ten times that of the next highest class. This is expressed as: log n(M) = a-bM, where n(M) is the number of earthquakes of a given magnitude M, and a- and b-values are constants representing the level of seismicity and the ratio of small to large events, respectively. In this formula, the expected maximum magnitude (c-value) in a given earthquake catalog is calculated using one more assumption: a maximum-magnitude earthquake should occur only once in a given period, because the c-value is not a characteristic parameter of the original GR formula. Utsu (1978) proposed that the GR formula be modified by introducing the c-value, and presented two formulas: a truncated GR formula (TGR), expressed as log n(M) = a - bM (M is equal to or smaller than c); n(M) = 0 (M is greater than c); and a modified GR formula (MGR), expressed as log n(M) = a - bM + log (c-M) (M is smaller than c); n(M) = 0 (M is equal to or greater than c). Calculations for 18 Japanese seismo-tectonic provinces revealed the following relation: b(GR) > b(TGR) > b(MGR). This is a theoretical relationship, which means that b- and c-values are relative parameters within one formula, and that comparison of b- and c-values between different GR formulas is meaningless. Furthermore, the distribution of b- and c-values in 18 intraplate seismo-tectonic provinces indicates that b- and c-values are very sensitive to the parameter Ms, which is the lower limit of magnitude above which the dataset is thought to be complete. Thus, it is necessary to ascertain that b- and c-values are stable with respect to Ms, and that c-values are appropriate to maximum magnitude when estimating these parameters within a given earthquake catalog and seismo-tectonic province. Reference: Kakimi, T., Matsuda, T, Kinugasa, Y., and Aida, I, 2002, A seismotectonic province map in and around the Japanese islands, submitted to Jour. Seimol. Soc. Japan (Zishin2). Ustu, T, 1978, Estimation of Parameters in Formulas for Frequency-Magnitude Relation of Earthquake Occurrence, Jour. Seimol. Soc. Japan (Zishin2), 31, 367-382.