Representations of the radiated energy in earthquakes

S U M M A R Y We investigate the representation of the radiated energy, E R, in earthquakes. In seismology E R is estimated from either far-field seismic waves or the stress and displacement on the fault plane. Although E R comes from the entire volume of the Earth, it can be expressed as an integral over the fault plane. However, the integrand cannot be given a simple physical meaning such as the radiated energy density on the fault plane. The stress on the fault plane changes rapidly during a seismic rupture. Although the energy radiated by this process is not included in the estimate of E R in a simplified practice in seismology, it is correctly included in the expression of E R in standard seismological practice. Using the representation theorem, we can express E R as a surface integral over the fault plane, with the integrand containing the slip function on the fault plane. However, the integrand at a point depends not only on the slip function at the point but also on the slip functions everywhere on the fault plane. Thus, the simple method in which E R is estimated by summation of the local energy flux on the fault plane does not yield a correct estimate.