Energy-based opinion on the correlation between stress drop and rupture speed

Stress drop $\Delta\tau$ and rupture speed $V_r$ are two important earthquake source parameters that control the characteristics of rupture process and the associated ground motion. However, how the two parameters correlate with one another is not fully clear and sometimes can be controversial. Here I propose using the energy balance relation derived from fracture mechanics to understand the correlation between $\Delta\tau$ and $V_r$. The central idea is to explore the balance between fracture energy $G_c$ and dynamic energy release rate $G_d$ (which itself is a function of $\Delta\tau$ and $V_r$) under various conditions of $G_c$. Using averaged values of $\Delta\tau$ and $V_r$ for comparison, it is shown that near constant $G_c$ can yield a positive correlation between $\Delta\tau$ and $V_r$, whereas variable $G_c$ dependent on space, time or other factors may cause a negative correlation between $\Delta\tau$ and $V_r$. These results suggest a need to examine the condition of other factors (such as $G_c$) when evaluating the correlation between $\Delta\tau$ and $V_r$. Extra issues can complicate the evaluation of the correlation between $\Delta\tau$ and $V_r$ when rupture process is inferred from far-field observations, accompanied by strong spatiotemporal variation, or followed by additional phases, which should be investigated by future studies.