Dependence of seismic energy on higher wavenumber components

Abstract:

Seismic Energy $E_S$ gives a minimum of strain energy drop defined as an inner product of spacial distribution of coseismic slip and stress change on a fault surface (Andrews 1978 JGR). Traditionally, $E_S$ has been obtained by multiplying mean stress drop and seismic moment divided by the rigidity by assuming the distribution of stress drop is constant in space, which yields an elliptic slip distribution. It has, however, been pointed out that slip distributions are approximated not as the elliptic distribution but as the $k$-squared model (Herrero & Bernard 1994 BSSA), so that the product of mean stress drop and seismic moment does not give proper estimation of $E_S$.

For the case of heterogeneous stress drop, the inner product requires shorter wavelength components of slip distribution (Andrews 1980 JGR). Mai & Beroza (2002 JGR) revealed that observed slip distributions in the wavenumber domain are well modeled with the von Karman power spectrum density parameterized by a corner wavenumber $k_c$ and the Hurst exponent $H$, and quantified these two parameters for some inversion results. Although they discussed a condition of convergence of the inner product, they did not consider dependence of $E_S$ on $k_c$, $H$, and a maximum wavenumber $k_{max}$. In this study, we analytically obtain the dependence and suggest how we should consider higher wavenumber components of slip distribution for estimation of $E_S$. We show that the relationship $E_S \propto C(k_{max}/k_c, H) \mu P^2 k_c^3$ holds, where $\mu$ is the rigidity, and $P$ is the seismic potency. An analytical solution of $C(k_{max}/k_c, H)$ tells us that even components of $k_{max}/k_c \sim 10$ or $100$ are not negligible for $E_S$ under $k$-squared model while such components do not contribute to $E_S$ for the elliptic slip distribution. We discuss this feature quantitatively and show some examples of estimation of $E_S$ based on results of slip inversions.