An Entropic Explanation for Gutenberg-Richter Scaling
Abstract
We develop a simple, entropic explanation for Gutenberg-Richter (G-R) size scaling of earthquakes on a single fault. We discretize the fault and consider all possible contiguous ruptures at that level of discretization. In this static model, we assume that slip scales with rupture length, and that the rupture rates at each point along the fault are consistent with an a priori long-term slip rate. These simple assumptions define an (underdetermined) non-negative least squares inverse problem. Each solution to this inverse problem is a set of earthquake rates that matches the slip-rate constraint. We use a Markov Chain Monte Carlo (MCMR) algorithm with boundary-condition mirroring to uniformly sample the solution space assuming constant slip rates along the fault. At finer discretizations, deviations from G-R behavior decrease, which is consistent with an entropic pressure towards G-R solutions. When the fault is discretized into 10 or more segments, random solutions found by the MCMR algorithm have G-R size scaling, even though there are trivial solutions that, for example, have earthquakes of only one size. This is because there are simply far more solutions that have G-R scaling; as the problem size increases, the strong degeneracy of G-R solutions results in other solutions becoming improbably rare. Also, the entropically favored G-R distribution has a b-value of approximately 1, which agrees with measured b-values in real earthquake catalogs. We also investigate more complex slip-rate distributions and find that the maximum-entropy size scaling is somewhat insensitive to the slip-rate data. The strong tendency toward power-law scaling can be broken by encouraging more small events by using “step-like” data that has large, sudden slip-rate variations along-strike. Smoothly varying the slip rates along-strike leads to solution sets that exhibit a strong degeneracy of G-R solutions with a b-value of approximately 1.
- Publication:
-
AGU Fall Meeting Abstracts
- Pub Date:
- December 2009
- Bibcode:
- 2009AGUFM.S43C..05P
- Keywords:
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- 3260 MATHEMATICAL GEOPHYSICS / Inverse theory;
- 7200 SEISMOLOGY;
- 7223 SEISMOLOGY / Earthquake interaction;
- forecasting;
- and prediction;
- 7260 SEISMOLOGY / Theory