1/f and the Earthquake Problem: Scaling constraints that facilitate operational earthquake forecasting

The difficulty of forecasting earthquakes can fundamentally be attributed to the self-similar, or "1/f", nature of seismic sequences. Specifically, the rate of occurrence of earthquakes is inversely proportional to their magnitude m, or more accurately to their scalar moment M. With respect to this "1/f problem," it can be argued that catalog selection (or equivalently, determining catalog constraints) constitutes the most significant challenge to seismicity based earthquake forecasting. Here, we address and introduce a potential solution to this most daunting problem. Specifically, we introduce a framework to constrain, or partition, an earthquake catalog (a study region) in order to resolve local seismicity. In particular, we combine Gutenberg-Richter (GR), rupture length, and Omori scaling with various empirical measurements to relate the size (spatial and temporal extents) of a study area (or bins within a study area) to the local earthquake magnitude potential - the magnitude of earthquake the region is expected to experience. From this, we introduce a new type of time dependent hazard map for which the tuning parameter space is nearly fully constrained. In a similar fashion, by combining various scaling relations and also by incorporating finite extents (rupture length, area, and duration) as constraints, we develop a method to estimate the Omori (temporal) and spatial aftershock decay parameters as a function of the parent earthquake's magnitude m. From this formulation, we develop an ETAS type model that overcomes many point-source limitations of contemporary ETAS. These models demonstrate promise with respect to earthquake forecasting applications. Moreover, the methods employed suggest a general framework whereby earthquake and other complex-system, 1/f type, problems can be constrained from scaling relations and finite extents.; Record-breaking hazard map of southern California, 2012-08-06. "Warm" colors indicate local acceleration (elevated hazard); "cool" colors indicate local Omori relaxation.