A new Bayesian formulation to locate earthquakes from body-wave direction of arrival
Résumé
Earthquake location is most of the time computed using arrival times of seismic waves observed on monitoring networks. However, the widespread use of three-component sensors gives also access to wave arrival direction, which can be used to better constrain the event hypocenter. This is well-known in single station monitoring or in hydraulic fracturing monitoring, where a string of three-component sensors is typically deployed in a single vertical well and where the P-wave polarization measurement is necessary to locate the induced seismicity. However, other seismic networks, especially with low coverage, could benefit from this available information.
We propose a new Bayesian formulation that defines a probability density function of the earthquake hypocenter as a function of the seismic body wave polarization. The likelihood function takes a three-component sensor perspective and uses the covariance matrix, which contains all necessary information including uncertainties, to quantify the polarization of the P- or the S-wave. It is consistent with directional statistics and compares a modeled arrival vector with the distribution observed at the sensor, assuming angular central Gaussian probabilities. This non-linear approach can also be used to orient three-component sensors from known seismic sources.
Unlike existing Gaussian formulations, the angular central Gaussian one does not reduce the polarization to a couple of azimuth and inclination angles and avoids approximate estimate of angular uncertainties. On the contrary, it replaces the polarization information in its spherical dataspace. As illustrated by synthetic tests in a 1D velocity model, the new formulation can lead to more confined locations and gives properly normalized probability density functions. Hence, earthquake hypocenters are provided with realistic uncertainty domains and the combination of the direction of arrival information with the arrival time information in a joint probability density function is straight forward.