Articles | Volume 7, issue 6
Research article
07 Nov 2016
Research article |  | 07 Nov 2016

Fully probabilistic seismic source inversion – Part 2: Modelling errors and station covariances

Simon C. Stähler and Karin Sigloch

Abstract. Seismic source inversion, a central task in seismology, is concerned with the estimation of earthquake source parameters and their uncertainties. Estimating uncertainties is particularly challenging because source inversion is a non-linear problem. In a companion paper, Stähler and Sigloch (2014) developed a method of fully Bayesian inference for source parameters, based on measurements of waveform cross-correlation between broadband, teleseismic body-wave observations and their modelled counterparts. This approach yields not only depth and moment tensor estimates but also source time functions.

A prerequisite for Bayesian inference is the proper characterisation of the noise afflicting the measurements, a problem we address here. We show that, for realistic broadband body-wave seismograms, the systematic error due to an incomplete physical model affects waveform misfits more strongly than random, ambient background noise. In this situation, the waveform cross-correlation coefficient CC, or rather its decorrelation D = 1 − CC, performs more robustly as a misfit criterion than ℓp norms, more commonly used as sample-by-sample measures of misfit based on distances between individual time samples.

From a set of over 900 user-supervised, deterministic earthquake source solutions treated as a quality-controlled reference, we derive the noise distribution on signal decorrelation D = 1 − CC of the broadband seismogram fits between observed and modelled waveforms. The noise on D is found to approximately follow a log-normal distribution, a fortunate fact that readily accommodates the formulation of an empirical likelihood function for D for our multivariate problem. The first and second moments of this multivariate distribution are shown to depend mostly on the signal-to-noise ratio (SNR) of the CC measurements and on the back-azimuthal distances of seismic stations. By identifying and quantifying this likelihood function, we make D and thus waveform cross-correlation measurements usable for fully probabilistic sampling strategies, in source inversion and related applications such as seismic tomography.

Short summary
Seismic source inversion is the method of inferring the spatial orientation of an earthquake source from seismic records. The results come with large uncertainties, which we try to estimate in a Bayesian approach. We propose an empirical relationship for the likelihood function based on a large dataset of deterministic solutions. This allows using the cross-correlation coefficient as a misfit criterion, which is better suited for waveform comparison than the popular root mean square or L2 norm.