The radiation of surface wave energy: Implications for volcanic tremor

Abstract:

The seismic energy radiated by active volcanism is one common measurement of eruption size. For example, the magnitudes of individual earthquakes in volcano-tectonic (VT) swarms can be summed and expressed in terms of cumulative magnitude, energy, or moment release. However, discrepancies exist in current practice when treating the radiated energy of volcano seismicity dominated by surface waves. This has implications for volcanic tremor, since eruption tremor typically originates at shallow depth and is made up of surface waves. In the absence of a method to compute surface wave energy, estimates of eruption energy partitioning between acoustic and seismic waves typically assume seismic energy is composed of body waves. Furthermore, without the proper treatment of surface wave energy, it is unclear how much volcanic tremor contributes to the overall seismic energy budget during volcanic unrest.

To address this issue, we derive, from first principles, the expression of surface wave radiated energy. In contrast with body waves, the surface wave energy equation is naturally expressed in the frequency domain instead of the time domain. We validate our result by reproducing an analytical solution for the radiated power of a vertical force source acting on a free surface. We further show that the surface wave energy equation leads to an explicit relationship between energy and the imaginary part of the surface wave Green’s tensor at the source location, a fundamental property recognized within the field of seismic interferometry.

With the new surface wave energy equation, we make clear connections to reduced displacement and propose an improved formula for the calculation of surface wave reduced displacement involving integration over the frequency band of tremor. As an alternative to reduced displacement, we show that reduced particle velocity squared is also a valid physical measure of tremor size, one based on seismic energy rate instead of seismic moment rate. These concepts are illustrated with data examples from recent eruptions of Pavlof and Okmok volcanoes in the Aleutian Arc.