Constitutive Law for Earthquake Production Based on Rate-and-State Friction: Theory and Application of Interacting Sources

The constitutive relationship of Dieterich (1994, https://doi.org/10.1029/93JB02581) for earthquake production has been applied widely to earthquake triggering. The Dieterich theory achieves mathematical simplicity and thus lends itself to practical applications through a number of simplifying assumptions. Perhaps the most suspect and criticized assumption is that seismic sources do not interact. Here I provide an extension of the constitutive framework, using the modified formulation of Heimisson and Segall (2018, https://doi.org/10.1029/2018JB015656) which accounts for source interactions. Moreover, I explore ways to account for earthquake magnitudes and highlight potential directions for future research. I identify conditions under which the constitutive relationship with interactions reduces to the noninteracting Dieterich theory, revealing that an interacting population may be modeled as an equivalent noninteracting population. This result may partly explain the success of the Dieterich theory in spite of assuming no interactions. These special conditions reveal that if the magnitude distribution (e.g., the b value) is altered by a stress perturbation, then the Dieterich model may either underpredict or overpredict the seismicity rate. Furthermore, the interaction model grants insight into the physical interpretation of the background seismicity rate, which is frequently considered to be a declustered rate. However, the results suggest that interactions do not change the total cumulative number of events over a long time span (t ≫ t_a). Finally, I apply the interaction theory to fault slip models, aftershock diffusion, and secondary triggering as illustrative examples.