Comparison of average stress drop measures for ruptures with heterogeneous stress change and implications for earthquake physics

Abstract

Stress drop, a measure of static stress change in earthquakes, is the subject of numerous investigations. Stress drop in an earthquake is likely to be spatially varying over the fault, creating a stress drop distribution. Representing this spatial distribution by a single number, as commonly done, implies averaging in space. In this study, we investigate similarities and differences between three different averages of the stress drop distribution used in earthquake studies. The first one, |$\overline{\Delta \sigma }_M$|⁠, is the commonly estimated stress drop based on the seismic moment and fault geometry/dimensions. It is known that |$\overline{\Delta \sigma }_M$| corresponds to averaging the stress drop distribution with the slip distribution due to uniform stress drop as the weighting function. The second one, |$\overline{\Delta \sigma }_A$|⁠, is the simplest (unweighted) average of the stress drop distribution over the fault, equal to the difference between the average stress levels on the fault before and after an earthquake. The third one, |$\overline{\Delta \sigma }_E$|⁠, enters discussions of energy partitioning and radiation efficiency; we show that it corresponds to averaging the stress drop distribution with the actual final slip at each point as the weighting function. The three averages, |$\overline{\Delta \sigma }_M$|⁠, |$\overline{\Delta \sigma }_A$|⁠, and |$\overline{\Delta \sigma }_E$|⁠, are often used interchangeably in earthquake studies and simply called ‘stress drop’. Yet they are equal to each other only for ruptures with spatially uniform stress drop, which results in an elliptical slip distribution for a circular rupture. Indeed, we find that other relatively simple slip shapes—such as triangular, trapezoidal or sinusoidal—already result in stress drop distributions with notable differences between |$\overline{\Delta \sigma }_M$|⁠, |$\overline{\Delta \sigma }_A$|⁠, and |$\overline{\Delta \sigma }_E$|⁠. Introduction of spatial slip heterogeneity results in further systematic differences between them, with |$\overline{\Delta \sigma }_E$| always being larger than |$\overline{\Delta \sigma }_M$|⁠, a fact that we have proven theoretically, and |$\overline{\Delta \sigma }_A$| almost always being the smallest. In particular, the value of the energy-related |$\overline{\Delta \sigma }_E$| significantly increases in comparison to the moment-based |$\overline{\Delta \sigma }_M$| with increasing roughness of the slip distribution over the fault. Previous studies used |$\overline{\Delta \sigma }_M$| in place of |$\overline{\Delta \sigma }_E$| in computing the radiation ratio ηR that compares the radiated energy in earthquakes to a characteristic part of their strain energy change. Typical values of ηR for large earthquakes were found to be from 0.25 to 1. Our finding that |$\overline{\Delta \sigma }_E \ge \overline{\Delta \sigma }_M$| allows us to interpret the values of ηR as the upper bound. We determine the restrictions placed by such estimates on the evolution of stress with slip at the earthquake source. We also find that |$\overline{\Delta \sigma }_E$| can be approximated by |$\overline{\Delta \sigma }_M$| if the latter is computed based on a reduced rupture area.