Probabilistic nonlinear waveform inversion method
The waveform inversion method we applied to determine the seismic moment tensor
has been developed by Wéber (2005,
2006,
2009).
It has already been successfully applied for estimating the full moment tensor
of both local and nearregional events in the Pannonian basin
(Wéber & Süle 2014,
Wéber 2016a,
Wéber 2016b).
The waveform inversion procedure works in the pointsource approximation.
A general seismic point source is described by six independent moment tensor rate
functions (MTRFs) allowing the moment tensor to vary arbitrarily as a function of time.
If the velocity structure and the event location are known, there is a linear
connection between the seismograms and the MTRFs. Basically, the MTRFs are obtained
by deconvolving the station specific Green’s functions (GFs) from the observed seismograms.
Especially in case of short epicentral distances, source mislocation can significantly bias
the results of any moment tensor inversion procedure. Therefore, we should further refine the
hypocentral coordinates using waveform data.
Treating the hypocentral coordinates as unknown parameters, however, leads to a nonlinear
inverse problem. The nonlinear waveform inversion procedure I have proposed to solve this problem
consists of the following main steps (see the flow diagram below):
 Step 1

The hypocenter of the event is estimated from observed arrival
times by a method that provides both the hypocentral coordinates
and their uncertainty. The
NonLinLoc software package or the
Bayesloc
multipleevent locator may be a suitable choice for this purpose.
 Step 2

Using the hypocenter distribution estimated in Step 1 as a priori
information and the observed waveforms as data, we perform a Bayesian inverse
calculation to improve the hypocentral location. For mapping the posterior
probability distribution (PPD) of the hypocenters, we apply the
octtree importance sampling algorithm
developed by Anthony Lomax and Andrew Curtis. As a result, we get a large
number of points that are samples from the PPD of the hypocenter. For all
these samples, the MTRFs are also calculated. Their distribution represents
the uncertainty of the MTRFs due to that of the source location.
 Step 3

Measurement errors and modeling errors also lead to MTRF uncertainty even for a
fixed source position. To estimate the overall uncertainties of the retrieved MTRFs,
we use a Monte Carlo simulation technique (Rubinstein & Kroese 2008).
The goal of Monte Carlo simulation is to determine how random variation in the input
data affects the uncertainty of the output. In our problem, source location and
seismograms represent the input data, whereas MTRFs are the output. In the course
of the simulation, we generate many new realizations of input data sets by randomly
generating new hypocenters and waveforms according to their respective distributions.
Then each generated input data set is inverted for MTRFs. The distribution of the
obtained set of solutions approximates well the PPD of the MTRFs.
After having mapped the PPD of the source location in the previous step,
generating a random hypocenter for the Monte Carlo simulation is straightforward.
For generating individual realizations of noisy seismograms, we first
calculate the waveform residual corresponding to the best MTRF solution
obtained in Step 2. We consider this residual as a realization of the
measurement and modeling errors. Convolving this error sample with a uniform
white noise yields a sample of simulated error: it differs from the original
sample of error but has the same amplitude spectrum. Then, we add the
simulated error to the observed seismograms and obtain a new realization
of waveforms which we invert for the MTRFs.
 Step 4

Assuming that the focal mechanism is constant in time, the previously obtained MTRFs
are decomposed into a timeinvariant moment tensor and a source time function (STF).
The problem is nonlinear and is solved by an iterative L1 norm minimization technique
(Wéber 2009).
To allow only forward slip during the rupture process, we impose a positivity constraint
on the STF.
After the decomposition of the MTRFs, a large number of moment tensor and STF solutions
are obtained that can be considered as samples from their respective PPDs.
The final estimates are given by the maximum likelihood points.
 Step 5

Once the moment tensors are retrieved, their principal axes are deduced.
Then each moment tensor is decomposed into an isotropic (ISO) part, representing an
explosive or implosive component, and into a deviatoric part, containing both
the doublecouple (DC) and the compensated linear vector dipole (CLVD) components.
The scalar seismic moments and percentages of the DC, CLVD and ISO components are
determined according to Vavrycuk (2015).
Finally, the distributions of the retrieved source parameters are displayed as
histogram plots.
The above described procedure can also be used when the STF is assumed
to be known. In that case, Monte Carlo simulation directly results in
samples from the posterior moment tensor distribution.
Acknowledgements
The reported investigation was financially supported by the
Hungarian Scientific Research Fund (Nos. T042572 and K68308).
References

Rubinstein, R.Y. & Kroese, D.P., 2008:
Simulation and the Monte Carlo Method,
John Wiley & Sons, Hoboken, New Jersey.

Vavrycuk, V., 2015:
Moment tensor decompositions revisited,
J. Seismol. 19, 231252.
