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

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

Symbols frequently used in this paper

Visual summary of the fully probabilistic source inversion algorithm
PRISM presented in the companion paper

The quantitative estimation of seismic source characteristics is one of the most important inverse problems in geophysics, from both scientific and societal points of views. Source parameters not only can be used to locate earthquakes and to understand earthquake mechanisms and their implications for tectonic settings and seismic hazard, but they are also important in seismic tomography, where accurate source information is a prerequisite for achieving optimal fits between observed and modelled (waveform) data.

Estimation of seismic source parameters includes an earthquake's location,
depth, fault plane and temporal rupture evolution. The inverse problem is
non-linear, and parameter correlations result in trade-offs and
non-uniqueness, e.g. the correlation between dip and scalar moment that was
discovered by

In a companion paper

The need for PRISM arose from our work in global-scale waveform tomography,
which fits broadband body-wave seismograms of moderate to large earthquakes
to modelled synthetics, up to the highest occurring frequencies
(

The required human supervision time called for full automatisation,
preferably in a Bayesian setting that would circumvent the occasional
divergence of the non-linear optimisation and would automatically diagnose
parameter trade-offs of the kind described. PRISM

To render ensemble sampling with the NA computationally feasible, the
dimensionality of the model parameter space has to be as small as possible,
preferably less than 20. Depth is one parameter, and a normalised description
of the moment tensor requires five more

This space is sampled by both stages of the neighbourhood algorithm,
resulting in an ensemble of source solutions

The primary measure of fit (or “input data”) for PRISM's source inversions
is the CC

Section

Section

To identify a likelihood function

Section

Three noise cases for compressional (

Bayesian inference estimates the posterior distribution

The exact formula for

“Good” solutions

The misfit functional has similar properties to a metric on

When the method of least squares is used to calculate the

In the case of a seismic waveform

For the estimation of the parameters

If each measurement

If the noise can be described well by the normal distribution, the

Hence,

Other authors have proposed to use misfits based on general

In summary, it is tempting to chose

These serious shortcomings motivate our proposal of alternate misfit criteria.

In a Bayesian context, the likelihood

For

With decreasing similarity of

In summary, the difference between a modelled

Next, we will test the signal decorrelation

Comparison of the

Distance between misfit value for the true source depth vs. the
plateau for depths 20–30 km in standard deviations. See
Fig.

We choose the signal decorrelation

For

If a time shift

Perturbation by convolution with a “modelling error function”

By adding a band-limited noise term

The right plot shows the value of the three waveform misfits

The

The

The cross-correlation misfit has the strongest difference between the plateau
of wrong depth solutions and the true one. For low noise levels, the minimum
is slightly wider than the one for the

Figure

Probability distribution of

In seismology, the cross-correlation coefficient

We present an empirical solution to this problem by drawing on a large,
pre-existing database of cross-correlation measurements that we assembled in
the context of deterministic source inversions, as described in Section 1.
Essentially we assert that our human expert knowledge and extensive
experience have generated a large, representative and highly
quality-controlled set of 900 teleseismic source parameter estimates that are
sufficiently close to the true source parameters to reveal the statistics of
the noise in the measurements

Our reasoning and procedure can be summed up as follows:

We can consider the measurements of misfit functional

In practice we never get to know

To evaluate the likelihood of a misfit value

The best we can do is to identify a suitable type of distribution and fit its parameters to the empirical histogram

The likelihood of a data vector

We will consider three candidate distributions for fitting an analytic

The beta and the exponential distributions are seen to overestimate the
number of very small

The log-normal distribution clearly yields the best approximation of the

The (univariate) log-normal distribution function is defined by two scale
parameters

The log-normal distribution also yields the best fit to our synthetic data
from Sect.

If random variable

The

Thus the use of

Colour shade map out a two-dimensional histogram of waveform
decorrelation

Here we describe how

This may be an oversimplification since ambient noise levels

To avoid this level of complexity, recall the investigation of
Sect.

SNR is defined as the integrated spectral energy in the signal time window,
divided by that of a 120 s noise window prior to the arrival of the
first body-wave energy. Signal time windows

Figure

By fitting functions of the form

Hence the log-normal distribution

The exact values for

Correlation in misfit between neighbouring stations. The measured
Pearson correlation (see Eq.

Decorrelation values

To check these systematics, we calculated the Pearson correlation coefficient

We need to adjust for the fact that stations

This permits comparison of

This azimuth-dependent correlation coefficient

An example of such a covariance matrix is shown in Fig.

Visualisation of an inter-station covariance matrix

Waveform amplitudes have not been considered so far, even though they provide
crucial constraints on focal mechanisms. Our amplitude measurement consists
of a comparison of the logarithmic energy content

Again our goal it to approximate the distribution of this misfit in order to
obtain an empirical likelihood function. The distribution of

In practice these concepts are integrated with the Bayesian source inversion
procedure of

For every new earthquake, download and archive a suitable selection of broadband, three-component, teleseismic seismograms (

Bandpass filter between 0.02 and 1.0 Hz. Rotate horizontal components to the RTZ system. Select signal time windows and noise time windows, and calculate SNR as defined in Eq. (

For each station, and for

Estimate correlation coefficient

Insert

For each source model

The most common approach to Bayesian inversion is to assert a simple noise
model for which an analytic likelihood function is known: this determines the
measure of misfit. We have gone the opposite route in designing a misfit

In fact, analytic probability densities are known for only a few misfit
functionals. By far the most commonly used are the Gaussian (normal)
distribution, associated with the

In practice however, the adoption of

More often than not, real data contain many more outliers than expected by
the normal distribution, certainly in the case of seismic data. Under the

The

Samples of real-world, band-limited time series are correlated. If a measured
seismogram of length

The situation is further complicated if the noise model can no longer be
purely additive (“

Another reason for leaving the Gaussian or

We are not sure whether there is a theoretical reason that the log-normal
distribution should be associated with the decorrelation misfit

As noted, the proposed empirical likelihood function

Other misfit criteria have been used in optimisation contexts in seismology.
For the purpose of source parameter inversion, their noise properties could
be investigated along the lines laid out by this work, and their empirical
likelihood functions studied. But unless their noise distributions turn out
to be as simple as for the

This paper presents an approach to Bayesian inference using the new misfit
criterion of waveform (de)correlation. Decorrelation

This opens the way for the methodically correct Bayesian sampling of
parameter estimation problems that use the cross-correlation CC or
decorrelation

The analysis has been performed on publicly available seismological data. All waveform data came from the IRIS and ORFEUS data management centres.

Simon C. Stähler conceived of the concept of the empirical likelihood in source inversion and did the data analysis. Karin Sigloch wrote the original source inversion code and created the earthquake database for the correlation misfit statistics. Both authors shared in the writing of the paper.

We thank M. Sambridge, R. Zhang, H. Igel and B. L. N. Kennett for fruitful discussions in an earlier stage of the work. T. Bodin and C. Tape helped improve the paper in the review process. Simon C. Stähler was supported by the Munich Centre of Advanced Computing (MAC) of the International Graduate School of Science and Engineering (IGSSE) at Technische Universität München. IGGSE also funded his research stay at the Research School for Earth Sciences at the Australian National University in Canberra, where part of this work was carried out. Karin Sigloch acknowledges funding by ERC Grant 639003 “DEEPTIME”, and Marie Curie CIG grant RHUM-RUM. This work was supported by the German Research Foundation (DFG) and the Technische Universität München within the funding programme Open Access Publishing. Edited by: C. Krawczyk Reviewed by: T. Bodin and C. Tape