relMT – Software to Determine Relative Seismic Moment Tensors Illustrated with Tectonic and Induced Seismicity

The seismic moment tensor (MT) delivers valuable information about the physical source process of a seismic event. It allows to distinguish between earthquakes, explosions and volcanic events, constraints the orientation of an earthquake rupture, and represents the most accurate estimate of the released seismic energy.

The computation of absolute MTs by waveform inversion is a data intensive task that is oftentimes feasible only for the largest events in a data set. Relative MTs rely on less subsurface information and may be computed for a large number of closely spaced weaker seismic events that are connected to an absolute MT through relative amplitude measurements. The relative MT method assumes that the Green’s functions between events is similar and that relative amplitudes are measured below the corner frequency of the largest event. Under these assumptions, the relative amplitude between seismograms can be attributed to the difference in moment tensor between events. Compared to absolute methods, path and site effects cancel out and do not need to be considered.

We here present relMT, an accessible, research-grade, open-source software package that facilitates computation of relative moment tensors for a large variety of data sets. The software takes as inputs seismic waveform, event locations, ray take-off angles, and a reference MT, as well as waveform headers and a configuration file. In synopsis, the similar waveforms are aligned to sub-sample accuracy under consideration of possible polarity reversals for P-waves and planar polarization of S-waves. Amplitude ratios between the aligned seismograms are measured on single seismic stations in a principal component framework. The relative amplitudes are combined mathematically with ray take-off angles, relative event distances and one absolute reference moment tensor in a linear system of equations. The solution of the equation system with algebraic methods yields all relative moment tensors at once. The uncertainty of the solutions is quantified using the bootstrap method. The software is under active development on GitHub (https://github.com/wasjabloch/relmt).

We illustrate the application of relMT using data sets of induced seismicity and tectonic aftershock seismicity. For the induced seismicity of the enhanced geothermal system in Helsinki, Finland, we are able to lower the magnitude threshold for which MTs can be computed from 0.5 to -0.5. For aftershocks in the Pamir highlands of Central Asia from 4.0 to 2.0.  For the data sets, this represents a 3- to 30-fold increase in the number of recovered MTs.