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, σ M , is the commonly estimated stress drop based on the seismic moment and fault geometry/dimensions. It is known that σ M corresponds to averaging the stress drop distribution with the slip distribution due to uniform stress drop as the weighting function. The second one, σ 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, σ 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, σ M , σ A , and σ 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 σ M , σ A , and σ E . Introduction of spatial slip heterogeneity results in further systematic differences between them, with σ E always being larger than σ M , a fact that we have proven theoretically, and σ A almost always being the smallest. In particular, the value of the energy-related σ E significantly increases in comparison to the moment-based σ M with increasing roughness of the slip distribution over the fault. Previous studies used σ M in place of σ E in computing the radiation ratio η R that compares the radiated energy in earthquakes to a characteristic part of their strain energy change. Typical values of η R for large earthquakes were found to be from 0.25 to 1. Our finding that σ E ≥ σ M allows us to interpret the values of η R 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 σ E can be approximated by σ M if the latter is computed based on a reduced rupture area.

New perspectives on self‐similarity for shallow thrust earthquakes

“…Classical slip inversions are performed on large fault planes, often rectangular and planar, to allow the slip to be contained within the parameter space, such that the area of nonzero slip is smaller than the area of the fault chosen for the inversion. To better estimate the area where slip occurred, a threshold of slip is defined to constrain the minimum of slip required to count each pixel in the area [Noda et al, 2013], which can yield trimming factors between the area of most slip and the total fault surface [Somerville et al, 1999;Noda et al, 2013;Ye et al, 2016aYe et al, , 2016b.…”

“…Noda et al [2013] differentiates three types of averaging, moment-based, area-based, and energy-based averaging, which agree if the stress drop is uniform on the fault. Note that the Noda et al [2013] study finds that moment-based stress drops likely underestimate the estimate of the average stress drop if the stress change is heterogeneous by up to a factor of 8. Brown et al [2015] confirms that stress drop varies strongly locally, and the choice of the threshold (trimming factor) especially affects total measures of stress drop.…”

The isolated ∼680 km deep 30 May 2015 MW 7.9 Ogasawara (Bonin) Islands earthquake

“…S8) and integrating it weighted by the slip distribution using the slipweighted stress method (Noda et al, 2013). σ is estimated as 38 MPa.…”

Energy Release of the 2013 M w 8.3 Sea of Okhotsk Earthquake and Deep Slab Stress Heterogeneity

“…Unfortunately, it is difficult in practice to determine the slip distribution in sufficient detail to estimate its spatial distribution. Numerical models 18 have shown that Δ . We use 0.15 as the trimming parameter for all of our finite-fault models for the mainshock and aftershock.…”