Reply to “Comment on ‘Models of Stochastic, Spatially Varying Stress in the Crust Compatible with Focal-Mechanism Data, and How Stress Inversions Can Be Biased toward the Stress Rate’ by Deborah

In her comment (Hardebeck, 2015) on our stress heterogeneity article (Smith and Heaton, 2011), Hardebeck suggests a different focal-mechanism error distribution than what we used in our 2011 article and suggests that this new error distribution will reduce our estimates of stress heterogeneity. In response to this, we have rerun our calculations three ways: (1) with the original mechanism error distribution from Smith and Heaton (2011), (2) with a mechanism error distribution similar to the one presented by Hardebeck (2015), and (3) with a mechanism error distribution derived from repeating earthquake statistics. We find the two new mechanism error models, relative to the original mechanism error distribution, reduce the heterogeneity ratio (HR) estimates by approximately 35%–40% (using Hardebeck’s suggested distribution) and by approximately 8%–10% (using the repeating earthquake based error distribution). Applying these two new mechanism error distribution models helps parameterize the estimates of stress heterogeneity amplitude but does not change the main novel points of the Smith and Heaton (2011) article. Namely, we find that focal-mechanism data are still compatible with a heterogeneous stress that is more dissimilar at large interevent distances and more correlated at small interevent distances and that a heterogeneous stress can bias traditional stress inversions toward the stressing rate function. Last, we demonstrate that as the size of the stress inversion region decreases and as the maximum variability of the heterogeneous stress decreases, the normalized stress inversion bias also decreases. This is consistent with taking the model of Smith and Heaton (2011) to the limit where region size decreases to a point source; however, most stress inversions may require dimensions closer to the outer scale of the stress (∼60 km for southern California) and hence experience significant stress inversion biasing toward the stressing rate. Hardebeck (2015) also refers to her 2010 article (Hardebeck, 2010) as a basis for refuting biasing of stress inversions. We disagree with Hardebeck’s stated refutation of stress inversion orientations in Smith and Heaton (2011) and Smith and Dieterich (2010); however, addressing the Hardebeck (2010) article is best accomplished via a direct modeling test in a separate comment that takes into account the complexities of the nucleation process, including rapidly evolving slip over the period of days.