Spatially Variable Stress in the Earth's Crust: Addressing Key Modeling Issues

Stress at some level in the Earth's crust is spatially variable, or else the crust would be extremely brittle - once a rupture began it wouldn't stop. The question is, "To what extent is stress spatially variable?" Specifically, what is the amplitude of the heterogeneity relative to the spatial mean, how does the relative amplitude vary over different length-scales, what is the time evolution of the stress heterogeneity, and how does the stress heterogeneity vary from one region to another? Smith and Heaton (2011) presented a fractal-like model of spatially varying stress in 3D and attempted to constrain its best-fitting statistical parameters for Southern California using focal mechanism data. Smith and Dieterich (2010) used a version of this model to generate a heterogeneous population of faults, which together with rate-state friction seismicity equations capture previously unmodeled effects of aftershock sequences. These models, which use spatially heterogeneous stress, have engendered discussion and debate about the degree and characterization of spatial heterogeneity. We present some of the key issues for spatially heterogeneous stress models, our response to the issues, and what we believe are outlying questions that need to be addressed. Ultimately, we argue for an intermediate model that is neither completely homogeneous (no spatial variation to the stress) nor highly heterogeneous (little to no spatial mean in the stress), but something in-between that takes into account spatial correlations. Such a model, consistent with the data-constrained Southern California stress parameters used in Smith and Dieterich (2010) and Smith and Heaton (2011), would have a maximum spatial variability (at the outer-scale), one to two times as large as the spatial mean. Appropriately characterizing spatially variable stress is critical for dynamic rupture calculations and hazard calculations. It has also been identified as a priority for the Southern California Earthquake Center, which is currently studying this via the Community Stress Model Working Group.