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felzer

HR: 12:05h
AN: S42C-08
TI: Calculating the Gutenberg-Richter b value
AU: * Felzer, K R
EM: kfelzer@gps.caltech.edu
AF: U. S. Geological Survey, 525 S. Wilson, Pasadena, CA 91106 United States
AB: The Gutenberg-Richter magnitude-frequency relationship for earthquakes is given by log(N(M)) = a - bM where M is magnitude, N(M) is the number of earthquakes of magnitude ≥ M, a is a constant, and b determines the relative number of earthquakes of different magnitudes. Seismic hazard analysis is very sensitive to the b value. If earthquake rates are based on the number of M ≥ 4 earthquakes, for example, a b value error on the order of 0.05 will cause the number of M ≥ 7 earthquakes forecast to be wrong by 40%. The value of b and its error is often miscalculated in practice, however. The common technique of solving for b with a least squares fit to the logarithm of the data, for example, leads to an answer that is biased for small data sets and apparent errors that are much smaller than the real ones. The maximum likelihood method of Aki (1965) is the most accurate way to calculate b, but large data sets are required. Monte Carlo simulations and equations from Aki (1965) indicate that a minimum of 2000 earthquakes are needed to calculate b to within 0.05 at 98% confidence. Also, as is well known, significant problems for b value calculation can be caused by choosing a minimum magnitude of completeness, M_C, that is too low. Often M_C is solved for by visual or quantitative inspection of logarithmic cumulative magnitude frequency plots for the point of maximum curvature. To test the adequacy of this technique I first use simulations to determine that unbiased b value solutions require that 95% of ≥ M_C earthquakes be present in the catalog. I then solve for b value in Southern California using a series of high M_C (≥ 2.5) and find b = 1.0. Next I find that the entire recorded magnitude distribution can be well fit by assuming a Gutenberg-Richter distribution with b=1 combined with a probability, P, that an earthquake will be detected and catalogued that varies with magnitude, M, as P(M) = 1 - C10^-M. The C value that best fits the Southern California catalog from 1995-2000 is 8.0, which corresponds to 95% of the earthquakes M ≥ 2.3 being recorded. Logarithmic cumulative magnitude frequency plots made from both real and simulated data, however, do not have any visible curvature at M 2.3. Instead, curvature only becomes clearly visible at around M 1.1, at which point 60% of the catalog is missing and calculated b values are low by 0.2. Even a more conservative visual pick of M_C = 1.5 corresponds to 25% data loss and b values low by 0.1. I also find that increasing magnitude error with decreasing magnitude in the California catalog creates significant problems for accurate b value solution. Given the large errors associated with b value calculation great care and exploration of all potential sources of error should be done before concluding that b values vary with time or space, and where there is not enough data to determine otherwise, a b value equal to the regional average should be assumed. There should also be great caution before assigning different b values or otherwise different magnitude distributions to earthquakes on and off of known major faults. Assignment of different magnitude distributions to large faults is often done in seismic hazard analysis, but there is scant evidence to support it. In particular along the Parkfield section of the San Andreas fault, where there is actually enough on-fault seismicity to make a robust measurement, careful assignment of M_C gives b=1.0, in agreement with the rest of California.
DE: 7223 Earthquake interaction, forecasting, and prediction (1217, 1242)
DE: 7230 Seismicity and tectonics (1207, 1217, 1240, 1242)
DE: 7290 Computational seismology
DE: 7299 General or miscellaneous
SC: Seismology [S]
MN: 2006 Fall Meeting